{"id":4158,"date":"2018-12-11T14:03:21","date_gmt":"2018-12-11T14:03:21","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/chapter\/evaluate-and-graph-logarithmic-functions\/"},"modified":"2020-05-17T06:06:25","modified_gmt":"2020-05-17T06:06:25","slug":"evaluate-and-graph-logarithmic-functions","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/chapter\/evaluate-and-graph-logarithmic-functions\/","title":{"raw":"Evaluate and Graph Logarithmic Functions","rendered":"Evaluate and Graph Logarithmic Functions"},"content":{"raw":"[latexpage]\n<div class=\"textbox textbox--learning-objectives\">\n<h3>Learning Objectives<\/h3>\nBy the end of this section, you will be able to:\n<ul>\n \t<li>Convert between exponential and logarithmic form<\/li>\n \t<li>Evaluate logarithmic functions<\/li>\n \t<li>Graph Logarithmic functions<\/li>\n \t<li>Solve logarithmic equations<\/li>\n \t<li>Use logarithmic models in applications<\/li>\n<\/ul>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167834120941\" class=\"be-prepared\">\n<p id=\"fs-id1167835415793\">Before you get started, take this readiness quiz.<\/p>\n\n<ol id=\"fs-id1167834396179\" type=\"1\">\n \t<li>Solve: \\({x}^{2}=81.\\)\n<div data-type=\"newline\"><\/div>\nIf you missed this problem, review <a href=\"\/contents\/da8478b4-93bc-4919-81a1-5e3267050e7e#fs-id1167836547919\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n \t<li>Evaluate: \\({3}^{-2}.\\)\n<div data-type=\"newline\"><\/div>\nIf you missed this problem, review <a href=\"\/contents\/3fa6a6c5-9a36-4dee-aea1-0166229f52fb#fs-id1167835527255\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n \t<li>Solve: \\({2}^{4}=3x-5.\\)\n<div data-type=\"newline\"><\/div>\nIf you missed this problem, review <a href=\"\/contents\/9f100e8f-6d15-4cae-bc22-c306e9d7d55c#fs-id1167836432956\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<\/ol>\n<\/div>\nWe have spent some time finding the inverse of many functions. It works well to \u2018undo\u2019 an operation with another operation. Subtracting \u2018undoes\u2019 addition, multiplication \u2018undoes\u2019 division, taking the square root \u2018undoes\u2019 squaring.\n<p id=\"fs-id1167826880213\">As we studied the exponential function, we saw that it is one-to-one as its graphs pass the horizontal line test. This means an exponential function does have an inverse. If we try our algebraic method for finding an inverse, we run into a problem.<\/p>\n<p id=\"fs-id1167835309061\">\\(\\begin{array}{cccc}\\begin{array}{}\\\\ \\\\ \\\\ \\text{Rewrite with}\\phantom{\\rule{0.2em}{0ex}}y=f\\left(x\\right).\\hfill \\\\ \\text{Interchange the variables}\\phantom{\\rule{0.2em}{0ex}}x\\phantom{\\rule{0.2em}{0ex}}\\text{and}\\phantom{\\rule{0.2em}{0ex}}y.\\hfill \\end{array}\\hfill &amp; &amp; &amp; \\begin{array}{ccc}\\hfill f\\left(x\\right)&amp; =\\hfill &amp; {a}^{x}\\hfill \\\\ \\hfill y&amp; =\\hfill &amp; {a}^{x}\\hfill \\\\ \\hfill x&amp; =\\hfill &amp; {a}^{y}\\hfill \\end{array}\\hfill \\\\ \\text{Solve for}\\phantom{\\rule{0.2em}{0ex}}y.\\hfill &amp; &amp; &amp; \\text{Oops! We have no way to solve for}\\phantom{\\rule{0.2em}{0ex}}y!\\hfill \\end{array}\\)<\/p>\n<p id=\"fs-id1167834214026\">To deal with this we define the logarithm function with base <em data-effect=\"italics\">a<\/em> to be the inverse of the exponential function \\(f\\left(x\\right)={a}^{x}.\\) We use the notation \\({f}^{-1}\\left(x\\right)={\\text{log}}_{a}x\\) and say the inverse function of the exponential function is the logarithmic function.<\/p>\n\n<div data-type=\"note\" id=\"fs-id1167831923783\">\n<div data-type=\"title\">Logarithmic Function<\/div>\n<p id=\"fs-id1167832198559\">The function \\(f\\left(x\\right)={\\text{log}}_{a}x\\) is the <strong data-effect=\"bold\">logarithmic function<\/strong> with base \\(a\\), where \\(a&gt;0,\\)\\(x&gt;0,\\) and \\(a\\ne 1.\\)<\/p>\n\n<div data-type=\"equation\" id=\"fs-id1167834061731\" class=\"unnumbered\" data-label=\"\">\\(y={\\text{log}}_{a}x\\phantom{\\rule{0.2em}{0ex}}\\text{is equivalent to}\\phantom{\\rule{0.2em}{0ex}}x={a}^{y}\\)<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167834185927\">\n<h3 data-type=\"title\">Convert Between Exponential and Logarithmic Form<\/h3>\nSince the equations \\(y={\\text{log}}_{a}x\\) and \\(x={a}^{y}\\) are equivalent, we can go back and forth between them. This will often be the method to solve some exponential and logarithmic equations. To help with converting back and forth let\u2019s take a close look at the equations. See <a href=\"#CNX_IntAlg_Figure_10_03_001\" class=\"autogenerated-content\">(Figure)<\/a>. Notice the positions of the exponent and base.\n<div class=\"bc-figure figure\" id=\"CNX_IntAlg_Figure_10_03_001\"><span data-type=\"media\" id=\"fs-id1167834505632\" data-alt=\"This figure shows the expression y equals log sub a of x, where y is the exponent and a is the base. Next to this expression we have x equals a to the y, where again y is the exponent and a is the base.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/jkdcbcintermediatealgebra\/wp-content\/uploads\/sites\/1021\/2018\/12\/CNX_IntAlg_Figure_10_03_001.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows the expression y equals log sub a of x, where y is the exponent and a is the base. Next to this expression we have x equals a to the y, where again y is the exponent and a is the base.\"><\/span><\/div>\n<p id=\"fs-id1167830904022\">If we realize the logarithm is the exponent it makes the conversion easier. You may want to repeat, \u201cbase to the exponent give us the number.\u201d<\/p>\n\n<div data-type=\"example\" id=\"fs-id1167834528356\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167834189130\">\n<p id=\"fs-id1167835365020\">Convert to logarithmic form: <span class=\"token\">\u24d0<\/span> \\({2}^{3}=8,\\) <span class=\"token\">\u24d1<\/span> \\({5}^{\\frac{1}{2}}=\\sqrt{5},\\) and <span class=\"token\">\u24d2<\/span> \\({\\left(\\frac{1}{2}\\right)}^{x}=\\frac{1}{16}.\\)<\/p>\n\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835479185\"><span data-type=\"media\" id=\"fs-id1167835240641\" data-alt=\"In part (a) we have 2 to the 3 power equals 8, where the 2 is red and the 3 is blue. Following this, we have blue y equals log sub red a of x. Then 3 equals log sub 2 of 8. Hence, if 2 cubed equals 8, then 3 equals log sub 2 of 8. In part (b) we have 5 to the 1 over 2 power equals square root of 5, where the 5 is red and the 1 over 2 is blue. Following this, we have blue y equals log sub red a of x. Then 1 over 2 equals log sub 5 of the square root of 5. Hence, if 5 to the 1 over 2 power equals the square root of 5, then 1 over 2 equals log sub 5 of the square root of 5. In part (c) we have 1 over 2 to the x power equals 1 over 16, where the 1 over 2 is red and the x is blue. Following this, we have blue y equals log sub red a of x. Then x equals log sub 1 over 2 of 1 over 16. Hence, if 1 over 2 to the x power equals 1 over 16, then x equals log sub 1 over 2 of 1 over 16.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/jkdcbcintermediatealgebra\/wp-content\/uploads\/sites\/1021\/2020\/05\/CNX_IntAlg_Figure_10_03_002_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"In part (a) we have 2 to the 3 power equals 8, where the 2 is red and the 3 is blue. Following this, we have blue y equals log sub red a of x. Then 3 equals log sub 2 of 8. Hence, if 2 cubed equals 8, then 3 equals log sub 2 of 8. In part (b) we have 5 to the 1 over 2 power equals square root of 5, where the 5 is red and the 1 over 2 is blue. Following this, we have blue y equals log sub red a of x. Then 1 over 2 equals log sub 5 of the square root of 5. Hence, if 5 to the 1 over 2 power equals the square root of 5, then 1 over 2 equals log sub 5 of the square root of 5. In part (c) we have 1 over 2 to the x power equals 1 over 16, where the 1 over 2 is red and the x is blue. Following this, we have blue y equals log sub red a of x. Then x equals log sub 1 over 2 of 1 over 16. Hence, if 1 over 2 to the x power equals 1 over 16, then x equals log sub 1 over 2 of 1 over 16.\"><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167831239572\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167830703207\">\n<div data-type=\"problem\" id=\"fs-id1167835420265\">\n<p id=\"fs-id1167832056505\">Convert to logarithmic form: <span class=\"token\">\u24d0<\/span> \\({3}^{2}=9\\) <span class=\"token\">\u24d1<\/span> \\({7}^{\\frac{1}{2}}=\\sqrt{7}\\) <span class=\"token\">\u24d2<\/span> \\({\\left(\\frac{1}{3}\\right)}^{x}=\\frac{1}{27}\\)<\/p>\n\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834395261\">\n<p id=\"fs-id1167831985714\"><span class=\"token\">\u24d0<\/span>\\({\\text{log}}_{3}9=2\\)<\/p>\n\n<div data-type=\"newline\"><\/div>\n<span class=\"token\">\u24d1<\/span>\\({\\text{log}}_{7}\\sqrt{7}=\\frac{1}{2}\\)<span class=\"token\">\u24d2<\/span>\\({\\text{log}}_{\\frac{1}{3}}\\frac{1}{27}=x\\)\n\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835362910\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835347825\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167835205986\">Convert to logarithmic form: <span class=\"token\">\u24d0<\/span> \\({4}^{3}=64\\) <span class=\"token\">\u24d1<\/span> \\({4}^{\\frac{1}{3}}=\\sqrt[3]{4}\\) <span class=\"token\">\u24d2<\/span> \\({\\left(\\frac{1}{2}\\right)}^{x}=\\frac{1}{32}\\)<\/p>\n\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835351385\">\n\n<span class=\"token\">\u24d0<\/span>\\({\\text{log}}_{4}64=3\\)\n<div data-type=\"newline\"><\/div>\n<span class=\"token\">\u24d1<\/span>\\({\\text{log}}_{4}\\sqrt[3]{4}=\\frac{1}{3}\\)<span class=\"token\">\u24d2<\/span>\\({\\text{log}}_{\\frac{1}{2}}\\frac{1}{32}=x\\)\n\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167831832871\">In the next example we do the reverse\u2014convert logarithmic form to exponential form.<\/p>\n\n<div data-type=\"example\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167831811617\">\n<div data-type=\"problem\" id=\"fs-id1167835233651\">\n<p id=\"fs-id1167835239496\">Convert to exponential form: <span class=\"token\">\u24d0<\/span> \\(2={\\text{log}}_{8}64,\\) <span class=\"token\">\u24d1<\/span> \\(0={\\text{log}}_{4}1,\\) and <span class=\"token\">\u24d2<\/span> \\(-3={\\text{log}}_{10}\\frac{1}{1000}.\\)<\/p>\n\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834064453\"><span data-type=\"media\" id=\"fs-id1167834195040\" data-alt=\"In part (a) we have 2 equals log sub 8 of 64, where the 2 is blue and the 8 is red. Following this, we have x equals red a to the blue y power. Then 64 equals 8 squared. Hence, if 2 equals log sub 8 of 64, then 64 equals 8 squared. In part (b) we have 0 equals log sub 4 of 1, where the 0 is blue and the 4 is red. Following this, we have x equals red a to the blue y power. Then 1 equals 4 to the zero power. Hence, if 0 equals log sub 4 of 1, then 1 equals 4 to the zero power. In part (c) we have negative 3 equals log sub 10 of 1 over 1000, where the negative 3 is blue and the 10 is red. Following this, we have x equals red a to the blue y power. Then 1 over 1000 equals 10 to the negative three power. Hence, if negative 3 equals log sub 10 of 1 over 1000, then 1 over 1000 equals 10 to the negative 3 power.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/jkdcbcintermediatealgebra\/wp-content\/uploads\/sites\/1021\/2020\/05\/CNX_IntAlg_Figure_10_03_003_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"In part (a) we have 2 equals log sub 8 of 64, where the 2 is blue and the 8 is red. Following this, we have x equals red a to the blue y power. Then 64 equals 8 squared. Hence, if 2 equals log sub 8 of 64, then 64 equals 8 squared. In part (b) we have 0 equals log sub 4 of 1, where the 0 is blue and the 4 is red. Following this, we have x equals red a to the blue y power. Then 1 equals 4 to the zero power. Hence, if 0 equals log sub 4 of 1, then 1 equals 4 to the zero power. In part (c) we have negative 3 equals log sub 10 of 1 over 1000, where the negative 3 is blue and the 10 is red. Following this, we have x equals red a to the blue y power. Then 1 over 1000 equals 10 to the negative three power. Hence, if negative 3 equals log sub 10 of 1 over 1000, then 1 over 1000 equals 10 to the negative 3 power.\"><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835530849\">\n<div data-type=\"problem\" id=\"fs-id1167832195683\">\n\nConvert to exponential form: <span class=\"token\">\u24d0<\/span> \\(3={\\text{log}}_{4}64\\) <span class=\"token\">\u24d1<\/span> \\(0={\\text{log}}_{x}1\\) <span class=\"token\">\u24d2<\/span> \\(-2={\\text{log}}_{10}\\frac{1}{100}\\)\n\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834225090\">\n<p id=\"fs-id1167826864312\"><span class=\"token\">\u24d0<\/span>\\(64={4}^{3}\\)<\/p>\n\n<div data-type=\"newline\"><\/div>\n<span class=\"token\">\u24d1<\/span>\\(1={x}^{0}\\)<span class=\"token\">\u24d2<\/span>\\(\\frac{1}{100}={10}^{-2}\\)\n\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167834188763\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167835302310\">Convert to exponential form: <span class=\"token\">\u24d0<\/span> \\(3={\\text{log}}_{3}27\\) <span class=\"token\">\u24d1<\/span> \\(0={\\text{log}}_{x}1\\) <span class=\"token\">\u24d2<\/span> \\(-1={\\text{log}}_{10}\\frac{1}{10}\\)<\/p>\n\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826986943\">\n<p id=\"fs-id1167834462881\"><span class=\"token\">\u24d0<\/span>\\(27={3}^{3}\\)<span class=\"token\">\u24d1<\/span>\\(1={x}^{0}\\)<\/p>\n\n<div data-type=\"newline\"><\/div>\n<span class=\"token\">\u24d2<\/span>\\(\\frac{1}{10}={10}^{-1}\\)\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167834537381\">\n<h3 data-type=\"title\">Evaluate Logarithmic Functions<\/h3>\n<p id=\"fs-id1167830837011\">We can solve and evaluate logarithmic equations by using the technique of converting the equation to its equivalent exponential equation.<\/p>\n\n<div data-type=\"example\" id=\"fs-id1167826978563\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167835218009\">\n<div data-type=\"problem\" id=\"fs-id1167834505593\">\n<p id=\"fs-id1167835306719\">Find the value of <em data-effect=\"italics\">x<\/em>: <span class=\"token\">\u24d0<\/span> \\({\\text{log}}_{x}36=2,\\) <span class=\"token\">\u24d1<\/span> \\({\\text{log}}_{4}x=3,\\) and <span class=\"token\">\u24d2<\/span> \\({\\text{log}}_{\\frac{1}{2}}\\frac{1}{8}=x.\\)<\/p>\n\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834431960\">\n<p id=\"fs-id1167834535355\"><span class=\"token\">\u24d0<\/span><\/p>\n\n<div data-type=\"newline\"><\/div>\n\\(\\begin{array}{cccc}&amp; &amp; &amp; \\phantom{\\rule{3.8em}{0ex}}{\\text{log}}_{x}36\\phantom{\\rule{0.2em}{0ex}}=\\phantom{\\rule{0.2em}{0ex}}2\\hfill \\\\ \\text{Convert to exponential form.}\\hfill &amp; &amp; &amp; \\phantom{\\rule{5.6em}{0ex}}{x}^{2}\\phantom{\\rule{0.2em}{0ex}}=\\phantom{\\rule{0.2em}{0ex}}36\\hfill \\\\ \\text{Solve the quadratic.}\\hfill &amp; &amp; &amp; \\phantom{\\rule{3.8em}{0ex}}x=6,\\phantom{\\rule{2em}{0ex}}\\overline{)x=-6}\\hfill \\\\ \\text{The base of a logarithmic function must be}\\hfill &amp; &amp; &amp; \\\\ \\text{positive, so we eliminate}\\phantom{\\rule{0.2em}{0ex}}x=-6.\\hfill &amp; &amp; &amp; \\phantom{\\rule{6.2em}{0ex}}x\\phantom{\\rule{0.2em}{0ex}}=\\phantom{\\rule{0.2em}{0ex}}6\\phantom{\\rule{2.5em}{0ex}}\\text{Therefore,}\\phantom{\\rule{0.2em}{0ex}}{\\text{log}}_{6}36=2.\\hfill \\end{array}\\)\n<p id=\"fs-id1167831040444\"><span class=\"token\">\u24d1<\/span><\/p>\n\n<div data-type=\"newline\"><\/div>\n\\(\\begin{array}{cccccc}&amp; &amp; &amp; \\phantom{\\rule{9.6em}{0ex}}{\\text{log}}_{4}x\\hfill &amp; =\\hfill &amp; 3\\hfill \\\\ \\text{Convert to exponential form.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{8.6em}{0ex}}{4}^{3}&amp; =\\hfill &amp; x\\hfill \\\\ \\text{Simplify.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{8.6em}{0ex}}x&amp; =\\hfill &amp; 64\\hfill &amp; \\phantom{\\rule{1.3em}{0ex}}\\text{Therefore,}\\phantom{\\rule{0.2em}{0ex}}{\\text{log}}_{4}64=3.\\hfill \\end{array}\\)\n<p id=\"fs-id1167828395816\"><span class=\"token\">\u24d2<\/span><\/p>\n\n<div data-type=\"newline\"><\/div>\n\\(\\begin{array}{}\\\\ \\\\ &amp; &amp; &amp; \\hfill \\phantom{\\rule{1em}{0ex}}{\\text{log}}_{\\frac{1}{2}}\\frac{1}{8}&amp; =\\hfill &amp; x\\hfill \\\\ \\text{Convert to exponential form.}\\hfill &amp; &amp; &amp; \\hfill {\\left(\\frac{1}{2}\\right)}^{x}&amp; =\\hfill &amp; \\frac{1}{8}\\hfill \\\\ \\text{Rewrite}\\phantom{\\rule{0.2em}{0ex}}\\frac{1}{8}\\phantom{\\rule{0.2em}{0ex}}\\text{as}\\phantom{\\rule{0.2em}{0ex}}{\\left(\\frac{1}{2}\\right)}^{3}.\\hfill &amp; &amp; &amp; \\hfill {\\left(\\frac{1}{2}\\right)}^{x}&amp; =\\hfill &amp; {\\left(\\frac{1}{2}\\right)}^{3}\\hfill \\\\ \\text{With the same base, the exponents must be equal.}\\hfill &amp; &amp; &amp; \\hfill x&amp; =\\hfill &amp; 3\\hfill &amp; \\text{Therefore,}\\phantom{\\rule{0.2em}{0ex}}{\\text{log}}_{\\frac{1}{2}}\\frac{1}{8}=3\\hfill \\end{array}\\)\n\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167832044030\">\n<div data-type=\"problem\" id=\"fs-id1167835216027\">\n<p id=\"fs-id1167835324805\">Find the value of \\(x:\\) <span class=\"token\">\u24d0<\/span> \\({\\text{log}}_{x}64=2\\) <span class=\"token\">\u24d1<\/span> \\({\\text{log}}_{5}x=3\\) <span class=\"token\">\u24d2<\/span> \\({\\text{log}}_{\\frac{1}{2}}\\frac{1}{4}=x\\)<\/p>\n\n<\/div>\n<div data-type=\"solution\">\n<div data-type=\"newline\"><\/div>\n<span class=\"token\">\u24d0<\/span>\\(x=8\\)<span class=\"token\">\u24d1<\/span>\\(x=125\\)<span class=\"token\">\u24d2<\/span>\\(x=2\\)\n\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167834099089\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835337372\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167835307439\">Find the value of \\(x:\\) <span class=\"token\">\u24d0<\/span> \\({\\text{log}}_{x}81=2\\) <span class=\"token\">\u24d1<\/span> \\({\\text{log}}_{3}x=5\\) <span class=\"token\">\u24d2<\/span> \\({\\text{log}}_{\\frac{1}{3}}\\frac{1}{27}=x\\)<\/p>\n\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835351129\">\n<div data-type=\"newline\"><\/div>\n<span class=\"token\">\u24d0<\/span>\n<div data-type=\"newline\"><\/div>\n\\(x=9\\)<span class=\"token\">\u24d1<\/span>\\(x=243\\)<span class=\"token\">\u24d2<\/span>\\(x=3\\)\n\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167830699532\">When see an expression such as \\({\\text{log}}_{3}27,\\) we can find its exact value two ways. By inspection we realize it means \\(\u201c3\\) to what power will be \\(27\u201d?\\) Since \\({3}^{3}=27,\\) we know \\({\\text{log}}_{3}27=3.\\) An alternate way is to set the expression equal to \\(x\\) and then convert it into an exponential equation.<\/p>\n\n<div data-type=\"example\" id=\"fs-id1167834501185\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167835534392\">\n<p id=\"fs-id1167835512014\">Find the exact value of each logarithm without using a calculator:<\/p>\n\n<div data-type=\"newline\"><\/div>\n<span class=\"token\">\u24d0<\/span> \\({\\text{log}}_{5}25,\\)\n<div data-type=\"newline\"><\/div>\n<span class=\"token\">\u24d1<\/span> \\({\\text{log}}_{9}3,\\) and <span class=\"token\">\u24d2<\/span> \\({\\text{log}}_{2}\\frac{1}{16}.\\)\n\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167832126125\">\n<p id=\"fs-id1167832052302\"><span class=\"token\">\u24d0<\/span><\/p>\n\n<div data-type=\"newline\"><\/div>\n\\(\\begin{array}{cccccc}&amp; &amp; &amp; \\hfill {\\text{log}}_{5}25&amp; &amp; \\\\ \\text{5 to what power will be}\\phantom{\\rule{0.2em}{0ex}}25?\\hfill &amp; &amp; &amp; \\hfill {\\text{log}}_{5}25&amp; =\\hfill &amp; 2\\hfill \\\\ \\text{Or}\\hfill &amp; &amp; &amp; &amp; \\\\ \\text{Set the expression equal to}\\phantom{\\rule{0.2em}{0ex}}x.\\hfill &amp; &amp; &amp; \\hfill {\\text{log}}_{5}25&amp; =\\hfill &amp; x\\hfill \\\\ \\text{Change to exponential form.}\\hfill &amp; &amp; &amp; \\hfill {5}^{x}&amp; =\\hfill &amp; 25\\hfill \\\\ \\text{Rewrite 25 as}\\phantom{\\rule{0.2em}{0ex}}{5}^{2}.\\hfill &amp; &amp; &amp; \\hfill {5}^{x}&amp; =\\hfill &amp; {5}^{2}\\hfill \\\\ \\text{With the same base the exponents must be equal.}\\hfill &amp; &amp; &amp; \\hfill x&amp; =\\hfill &amp; 2\\hfill &amp; \\text{Therefore,}\\phantom{\\rule{0.2em}{0ex}}{\\text{log}}_{5}25=2.\\hfill \\end{array}\\)\n\n<span class=\"token\">\u24d1<\/span>\n<div data-type=\"newline\"><\/div>\n\\(\\begin{array}{cccccc}&amp; &amp; &amp; \\hfill {\\text{log}}_{9}3&amp; &amp; \\\\ \\text{Set the expression equal to}\\phantom{\\rule{0.2em}{0ex}}x.\\hfill &amp; &amp; &amp; \\hfill {\\text{log}}_{9}3&amp; =\\hfill &amp; x\\hfill \\\\ \\text{Change to exponential form.}\\hfill &amp; &amp; &amp; \\hfill {9}^{x}&amp; =\\hfill &amp; 3\\hfill \\\\ \\text{Rewrite 9 as}\\phantom{\\rule{0.2em}{0ex}}{3}^{2}.\\hfill &amp; &amp; &amp; \\hfill {\\left({3}^{2}\\right)}^{x}&amp; =\\hfill &amp; {3}^{1}\\hfill \\\\ \\text{Simplify the exponents.}\\hfill &amp; &amp; &amp; \\hfill {3}^{2x}&amp; =\\hfill &amp; {3}^{1}\\hfill \\\\ \\text{With the same base the exponents must be equal.}\\hfill &amp; &amp; &amp; \\hfill 2x&amp; =\\hfill &amp; 1\\hfill \\\\ \\text{Solve the equation.}\\hfill &amp; &amp; &amp; \\hfill x&amp; =\\hfill &amp; \\frac{1}{2}\\hfill &amp; \\text{Therefore,}\\phantom{\\rule{0.2em}{0ex}}{\\text{log}}_{9}3=\\frac{1}{2}.\\hfill \\end{array}\\)\n<p id=\"fs-id1167834593440\"><span class=\"token\">\u24d2<\/span><\/p>\n\n<div data-type=\"newline\"><\/div>\n\\(\\begin{array}{cccccc}&amp; &amp; &amp; \\hfill {\\text{log}}_{2}\\frac{1}{16}&amp; &amp; \\\\ \\text{Set the expression equal to}\\phantom{\\rule{0.2em}{0ex}}x.\\hfill &amp; &amp; &amp; \\hfill {\\text{log}}_{2}\\frac{1}{16}&amp; =\\hfill &amp; x\\hfill \\\\ \\text{Change to exponential form.}\\hfill &amp; &amp; &amp; \\hfill {2}^{x}&amp; =\\hfill &amp; \\frac{1}{16}\\hfill \\\\ \\text{Rewrite 16 as}\\phantom{\\rule{0.2em}{0ex}}{2}^{4}.\\hfill &amp; &amp; &amp; \\hfill {2}^{x}&amp; =\\hfill &amp; \\frac{1}{{2}^{4}}\\hfill \\\\ &amp; &amp; &amp; \\hfill {2}^{x}&amp; =\\hfill &amp; {2}^{-4}\\hfill \\\\ \\text{With the same base the exponents must be equal.}\\hfill &amp; &amp; &amp; \\hfill x&amp; =\\hfill &amp; -4\\hfill &amp; \\text{Therefore,}\\phantom{\\rule{0.2em}{0ex}}{\\text{log}}_{2}\\frac{1}{16}=-4.\\hfill \\end{array}\\)\n\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" class=\"try\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167834377442\">\n<p id=\"fs-id1167831823950\">Find the exact value of each logarithm without using a calculator:<\/p>\n\n<div data-type=\"newline\"><\/div>\n<span class=\"token\">\u24d0<\/span> \\({\\text{log}}_{12}144\\)\n<div data-type=\"newline\"><\/div>\n<span class=\"token\">\u24d1<\/span> \\({\\text{log}}_{4}2\\)\n<div data-type=\"newline\"><\/div>\n<span class=\"token\">\u24d2<\/span> \\({\\text{log}}_{2}\\frac{1}{32}\\)\n\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834556149\">\n<div data-type=\"newline\"><\/div>\n<span class=\"token\">\u24d0<\/span>\n<div data-type=\"newline\"><\/div>\n2 <span class=\"token\">\u24d1<\/span> \\(\\frac{1}{2}\\) <span class=\"token\">\u24d2<\/span> \\(-5\\)\n\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167834346951\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167834464088\">\n<div data-type=\"problem\" id=\"fs-id1167834299816\">\n<p id=\"fs-id1167834547189\">Find the exact value of each logarithm without using a calculator:<\/p>\n\n<div data-type=\"newline\"><\/div>\n<span class=\"token\">\u24d0<\/span> \\({\\text{log}}_{9}81\\)\n<div data-type=\"newline\"><\/div>\n<span class=\"token\">\u24d1<\/span> \\({\\text{log}}_{8}2\\)\n<div data-type=\"newline\"><\/div>\n<span class=\"token\">\u24d2<\/span> \\({\\text{log}}_{3}\\frac{1}{9}\\)\n\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834537147\">\n\n<span class=\"token\">\u24d0<\/span> 2 <span class=\"token\">\u24d1<\/span> \\(\\frac{1}{3}\\) <span class=\"token\">\u24d2<\/span> \\(-2\\)\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167835233646\">\n<h3 data-type=\"title\">Graph Logarithmic Functions<\/h3>\n<p id=\"fs-id1167835306032\">To graph a logarithmic function \\(y={\\text{log}}_{a}x,\\) it is easiest to convert the equation to its exponential form, \\(x={a}^{y}.\\) Generally, when we look for ordered pairs for the graph of a function, we usually choose an <em data-effect=\"italics\">x<\/em>-value and then determine its corresponding <em data-effect=\"italics\">y<\/em>-value. In this case you may find it easier to choose <em data-effect=\"italics\">y<\/em>-values and then determine its corresponding <em data-effect=\"italics\">x<\/em>-value.<\/p>\n\n<div data-type=\"example\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167832059991\">\n<div data-type=\"problem\" id=\"fs-id1167831071344\">\n<p id=\"fs-id1167832066288\">Graph \\(y={\\text{log}}_{2}x.\\)<\/p>\n\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826849526\">\n<p id=\"fs-id1167831239155\">To graph the function, we will first rewrite the logarithmic equation, \\(y={\\text{log}}_{2}x,\\) in exponential form, \\({2}^{y}=x.\\)<\/p>\n<p id=\"fs-id1167826994962\">We will use point plotting to graph the function. It will be easier to start with values of <em data-effect=\"italics\">y<\/em> and then get <em data-effect=\"italics\">x<\/em>.<\/p>\n\n<table class=\"unnumbered\" summary=\"This table has three columns and seven rows. The first row is a header row and it reads y, 2 to the y power equals x, and (x, y). In the first column below y we have negative 2, negative 1, 0, 1, 2, and 3. In the second column below 2 to the y power equals x we have 2 to the negative 2 power equals 1 over 2 squared which equals 1 over 4, 2 to the negative 1 power equals 1 over 2 to the first power which equals 1 over 2, 2 to the negative 0 power equals 2, 2 to the 1 power equals 2, 2 squared equals 4, and 2 cubed equals 8. In the third column below (x, y) we have (1 over 4, 2), (1 over 2, negative 1), (1, 0), (2, 1), (4, 2), and (8, 3).\" data-label=\"\">\n<thead>\n<tr>\n<th data-valign=\"middle\" data-align=\"left\">\\(y\\)<\/th>\n<th data-valign=\"middle\" data-align=\"left\">\\({2}^{y}=x\\)<\/th>\n<th data-valign=\"middle\" data-align=\"left\">\\(\\left(x,y\\right)\\)<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td data-valign=\"middle\" data-align=\"left\">\\(-2\\)<\/td>\n<td data-valign=\"middle\" data-align=\"left\">\\({2}^{-2}=\\frac{1}{{2}^{2}}=\\frac{1}{4}\\)<\/td>\n<td data-valign=\"middle\" data-align=\"left\">\\(\\left(\\frac{1}{4},2\\right)\\)<\/td>\n<\/tr>\n<tr>\n<td data-valign=\"middle\" data-align=\"left\">\\(-1\\)<\/td>\n<td data-valign=\"middle\" data-align=\"left\">\\({2}^{-1}=\\frac{1}{{2}^{1}}=\\frac{1}{2}\\)<\/td>\n<td data-valign=\"middle\" data-align=\"left\">\\(\\left(\\frac{1}{2},-1\\right)\\)<\/td>\n<\/tr>\n<tr>\n<td data-valign=\"middle\" data-align=\"left\">0<\/td>\n<td data-valign=\"middle\" data-align=\"left\">\\({2}^{0}=1\\)<\/td>\n<td data-valign=\"middle\" data-align=\"left\">\\(\\left(1,0\\right)\\)<\/td>\n<\/tr>\n<tr>\n<td data-valign=\"middle\" data-align=\"left\">1<\/td>\n<td data-valign=\"middle\" data-align=\"left\">\\({2}^{1}=2\\)<\/td>\n<td data-valign=\"middle\" data-align=\"left\">\\(\\left(2,1\\right)\\)<\/td>\n<\/tr>\n<tr>\n<td data-valign=\"middle\" data-align=\"left\">2<\/td>\n<td data-valign=\"middle\" data-align=\"left\">\\({2}^{2}=4\\)<\/td>\n<td data-valign=\"middle\" data-align=\"left\">\\(\\left(4,2\\right)\\)<\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">3<\/td>\n<td data-valign=\"middle\" data-align=\"left\">\\({2}^{3}=8\\)<\/td>\n<td data-valign=\"top\" data-align=\"left\">\\(\\left(8,3\\right)\\)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<span data-type=\"media\" id=\"fs-id1167835328463\" data-alt=\"This figure shows the logarithmic curve going through the points (1 over 2, negative 1), (1, 0), and (2, 1).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/jkdcbcintermediatealgebra\/wp-content\/uploads\/sites\/1021\/2020\/05\/CNX_IntAlg_Figure_10_03_004_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows the logarithmic curve going through the points (1 over 2, negative 1), (1, 0), and (2, 1).\"><\/span>\n\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835201098\" class=\"try\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167831116958\">Graph: \\(y={\\text{log}}_{3}x.\\)<\/p>\n\n<\/div>\n<div data-type=\"solution\">\n<div data-type=\"newline\"><\/div>\n<span data-type=\"media\" data-alt=\"This figure shows the logarithmic curve going through the points (1 over 3, negative 1), (1, 0), and (3, 1).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/jkdcbcintermediatealgebra\/wp-content\/uploads\/sites\/1021\/2020\/05\/CNX_IntAlg_Figure_10_03_301_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows the logarithmic curve going through the points (1 over 3, negative 1), (1, 0), and (3, 1).\"><\/span>\n\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" class=\"try\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167831117010\">Graph: \\(y={\\text{log}}_{5}x.\\)<\/p>\n\n<\/div>\n<div data-type=\"solution\">\n<div data-type=\"newline\"><\/div>\n<span data-type=\"media\" id=\"fs-id1167835336536\" data-alt=\"This figure shows the logarithmic curve going through the points (1 over 5, negative 1), (1, 0), and (5, 1).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/jkdcbcintermediatealgebra\/wp-content\/uploads\/sites\/1021\/2020\/05\/CNX_IntAlg_Figure_10_03_302_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows the logarithmic curve going through the points (1 over 5, negative 1), (1, 0), and (5, 1).\"><\/span>\n\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167831191366\">The graphs of \\(y={\\text{log}}_{2}x,\\)\\(y={\\text{log}}_{3}x,\\) and \\(y={\\text{log}}_{5}x\\) are the shape we expect from a logarithmic function where \\(a&gt;1.\\)<\/p>\n<p id=\"fs-id1167835361417\">We notice that for each function the graph contains the point \\(\\left(1,0\\right).\\) This make sense because \\(0={\\text{log}}_{a}1\\) means \\({a}^{0}=1\\) which is true for any <em data-effect=\"italics\">a<\/em>.<\/p>\n<p id=\"fs-id1167834527903\">The graph of each function, also contains the point \\(\\left(a,1\\right).\\) This makes sense as \\(1={\\text{log}}_{a}a\\) means \\({a}^{1}=a.\\) which is true for any <em data-effect=\"italics\">a<\/em>.<\/p>\n<p id=\"fs-id1167835173724\">Notice too, the graph of each function \\(y={\\text{log}}_{a}x\\) also contains the point \\(\\left(\\frac{1}{a},-1\\right).\\) This makes sense as \\(-1={\\text{log}}_{a}\\frac{1}{a}\\) means \\({a}^{-1}=\\frac{1}{a},\\) which is true for any <em data-effect=\"italics\">a<\/em>.<\/p>\n<p id=\"fs-id1167835308174\">Look at each graph again. Now we will see that many characteristics of the logarithm function are simply \u2019mirror images\u2019 of the characteristics of the corresponding exponential function.<\/p>\nWhat is the domain of the function? The graph never hits the <em data-effect=\"italics\">y<\/em>-axis. The domain is all positive numbers. We write the domain in interval notation as \\(\\left(0,\\infty \\right).\\)\n<p id=\"fs-id1167826804614\">What is the range for each function? From the graphs we can see that the range is the set of all real numbers. There is no restriction on the range. We write the range in interval notation as \\(\\left(\\text{\u2212}\\infty ,\\infty \\right).\\)<\/p>\n<p id=\"fs-id1167835423129\">When the graph approaches the <em data-effect=\"italics\">y<\/em>-axis so very closely but will never cross it, we call the line \\(x=0,\\) the <em data-effect=\"italics\">y<\/em>-axis, a vertical asymptote.<\/p>\n\n<div data-type=\"note\" id=\"fs-id1167830705415\">\n<div data-type=\"title\">Properties of the Graph of \\(y={\\text{log}}_{a}x\\) when \\(a&gt;1\\)<\/div>\n<table class=\"unnumbered\" summary=\"Table has two columns and six rows. The first row shows the domain is 0 to infinity. The second row shows the range is negative infinity to infinity. The third row shows the x intercept is 1, 0. The fourth row shows there is no y-intercept. The fifth row shows the function contains a, 1 and 1 over a, negative 1. The sixth column shows the asymptote is the y axis.\" data-label=\"\">\n<tbody>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Domain<\/td>\n<td data-valign=\"top\" data-align=\"left\">\\(\\left(0,\\infty \\right)\\)<\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Range<\/td>\n<td data-valign=\"top\" data-align=\"left\">\\(\\left(\\text{\u2212}\\infty ,\\infty \\right)\\)<\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">\\(x\\text{-}\\text{intercept}\\)<\/td>\n<td data-valign=\"top\" data-align=\"left\">\\(\\left(1,0\\right)\\)<\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">\\(y\\text{-}\\text{intercept}\\)<\/td>\n<td data-valign=\"top\" data-align=\"left\">None<\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Contains<\/td>\n<td data-valign=\"top\" data-align=\"left\">\\(\\left(a,1\\right),\\)\\(\\left(\\frac{1}{a},-1\\right)\\)<\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Asymptote<\/td>\n<td data-valign=\"top\" data-align=\"left\">\\(y\\text{-}\\text{axis}\\)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<span data-type=\"media\" data-alt=\"This figure shows the logarithmic curve going through the points (1 over a, negative 1), (1, 0), and (a, 1).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/jkdcbcintermediatealgebra\/wp-content\/uploads\/sites\/1021\/2020\/05\/CNX_IntAlg_Figure_10_03_005_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows the logarithmic curve going through the points (1 over a, negative 1), (1, 0), and (a, 1).\"><\/span>\n\n<\/div>\nOur next example looks at the graph of \\(y={\\text{log}}_{a}x\\) when \\(0&lt;a&lt;1.\\)\n<div data-type=\"example\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167835517856\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167835230729\">Graph \\(y={\\text{log}}_{\\frac{1}{3}}x.\\)<\/p>\n\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834059251\">\n\nTo graph the function, we will first rewrite the logarithmic equation, \\(y={\\text{log}}_{\\frac{1}{3}}x,\\) in exponential form, \\({\\left(\\frac{1}{3}\\right)}^{y}=x.\\)\n<p id=\"fs-id1167834229268\">We will use point plotting to graph the function. It will be easier to start with values of <em data-effect=\"italics\">y<\/em> and then get <em data-effect=\"italics\">x<\/em>.<\/p>\n\n<table class=\"unnumbered\" summary=\"This table has three columns and seven rows. The first row is a header row and it reads y, 1 over 3 to the y power equals x and (x, y). In the first column below y, we have negative 2, negative 1, 0, 1, 2, and 3. In the second column below 1 over 3 to the y power equals x we have 1 over 3 to the negative 2 power equals 3 squared which equals 9, 1 over 3 to the negative 1 power equals 3 to the first power which equals 3, 1 over 3 to the 0 power equals 1, 1 over 3 to the 1 power equals 1 over 3, 1 over 3 squared equals 1 over 9, and 1 over 3 cubed equals 1 over 27. In the third column below (x, y) we have (9, negative 2), (3, negative 1), (1, 0), (1 over 3, 1), (1 over 9, 2), and (1 over 27, 3).\" data-label=\"\">\n<thead>\n<tr>\n<th data-valign=\"middle\" data-align=\"left\">\\(y\\)<\/th>\n<th data-valign=\"middle\" data-align=\"left\">\\({\\left(\\frac{1}{3}\\right)}^{y}=x\\)<\/th>\n<th data-valign=\"middle\" data-align=\"left\">\\(\\left(x,y\\right)\\)<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td data-valign=\"middle\" data-align=\"left\">\\(-2\\)<\/td>\n<td data-valign=\"middle\" data-align=\"left\">\\({\\left(\\frac{1}{3}\\right)}^{-2}={3}^{2}=9\\)<\/td>\n<td data-valign=\"middle\" data-align=\"left\">\\(\\left(9,-2\\right)\\)<\/td>\n<\/tr>\n<tr>\n<td data-valign=\"middle\" data-align=\"left\">\\(-1\\)<\/td>\n<td data-valign=\"middle\" data-align=\"left\">\\({\\left(\\frac{1}{3}\\right)}^{-1}={3}^{1}=3\\)<\/td>\n<td data-valign=\"middle\" data-align=\"left\">\\(\\left(3,-1\\right)\\)<\/td>\n<\/tr>\n<tr>\n<td data-valign=\"middle\" data-align=\"left\">0<\/td>\n<td data-valign=\"middle\" data-align=\"left\">\\({\\left(\\frac{1}{3}\\right)}^{0}=1\\)<\/td>\n<td data-valign=\"middle\" data-align=\"left\">\\(\\left(1,0\\right)\\)<\/td>\n<\/tr>\n<tr>\n<td data-valign=\"middle\" data-align=\"left\">1<\/td>\n<td data-valign=\"middle\" data-align=\"left\">\\({\\left(\\frac{1}{3}\\right)}^{1}=\\frac{1}{3}\\)<\/td>\n<td data-valign=\"middle\" data-align=\"left\">\\(\\left(\\frac{1}{3},1\\right)\\)<\/td>\n<\/tr>\n<tr>\n<td data-valign=\"middle\" data-align=\"left\">2<\/td>\n<td data-valign=\"middle\" data-align=\"left\">\\({\\left(\\frac{1}{3}\\right)}^{2}=\\frac{1}{9}\\)<\/td>\n<td data-valign=\"middle\" data-align=\"left\">\\(\\left(\\frac{1}{9},2\\right)\\)<\/td>\n<\/tr>\n<tr>\n<td data-valign=\"middle\" data-align=\"left\">3<\/td>\n<td data-valign=\"middle\" data-align=\"left\">\\({\\left(\\frac{1}{3}\\right)}^{3}=\\frac{1}{27}\\)<\/td>\n<td data-valign=\"top\" data-align=\"left\">\\(\\left(\\frac{1}{27},3\\right)\\)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<span data-type=\"media\" id=\"fs-id1167831102974\" data-alt=\"This figure shows the logarithmic curve going through the points (1 over 3, 1), (1, 0), and (3, negative 1).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/jkdcbcintermediatealgebra\/wp-content\/uploads\/sites\/1021\/2020\/05\/CNX_IntAlg_Figure_10_03_006_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows the logarithmic curve going through the points (1 over 3, 1), (1, 0), and (3, negative 1).\"><\/span>\n\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167834091580\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167834534505\">\n<div data-type=\"problem\" id=\"fs-id1167826940840\">\n\nGraph: \\(y={\\text{log}}_{\\frac{1}{2}}x.\\)\n\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835380353\">\n<div data-type=\"newline\"><\/div>\n<span data-type=\"media\" data-alt=\"This figure shows the logarithmic curve going through the points (1 over 2, 1), (1, 0), and (2, negative 1).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/jkdcbcintermediatealgebra\/wp-content\/uploads\/sites\/1021\/2020\/05\/CNX_IntAlg_Figure_10_03_303_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows the logarithmic curve going through the points (1 over 2, 1), (1, 0), and (2, negative 1).\"><\/span>\n<p id=\"fs-id1167835340700\"><\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167832059450\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167834429351\">\n<div data-type=\"problem\" id=\"fs-id1167835264420\">\n<p id=\"fs-id1167835218014\">Graph: \\(y={\\text{log}}_{\\frac{1}{4}}x.\\)<\/p>\n\n<\/div>\n<div data-type=\"solution\">\n<div data-type=\"newline\"><\/div>\n<span data-type=\"media\" id=\"fs-id1167831103870\" data-alt=\"This figure shows the logarithmic curve going through the points (1 over 4, 1), (1, 0), and (4, negative 1).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/jkdcbcintermediatealgebra\/wp-content\/uploads\/sites\/1021\/2020\/05\/CNX_IntAlg_Figure_10_03_304_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows the logarithmic curve going through the points (1 over 4, 1), (1, 0), and (4, negative 1).\"><\/span>\n\n<\/div>\n<\/div>\n<\/div>\nNow, let\u2019s look at the graphs \\(y={\\text{log}}_{\\frac{1}{2}}x,\\phantom{\\rule{0.2em}{0ex}}y={\\text{log}}_{\\frac{1}{3}}x\\) and \\(y={\\text{log}}_{\\frac{1}{4}}x\\), so we can identify some of the properties of logarithmic functions where \\(0&lt;a&lt;1.\\)\n<p id=\"fs-id1167834516390\">The graphs of all have the same basic shape. While this is the shape we expect from a logarithmic function where \\(0&lt;a&lt;1.\\)<\/p>\n<p id=\"fs-id1167835595491\">We notice, that for each function again, the graph contains the points,\\(\\left(1,0\\right),\\)\\(\\left(a,1\\right),\\)\\(\\left(\\frac{1}{a},-1\\right).\\) This make sense for the same reasons we argued above.<\/p>\n<p id=\"fs-id1167835381440\">We notice the domain and range are also the same\u2014the domain is \\(\\left(0,\\infty \\right)\\) and the range is \\(\\left(\\text{\u2212}\\infty ,\\infty \\right).\\) The \\(y\\)-axis is again the vertical asymptote.<\/p>\n<p id=\"fs-id1167828396212\">We will summarize these properties in the chart below. Which also include when \\(a&gt;1.\\)<\/p>\n\n<div data-type=\"note\" id=\"fs-id1167835233819\">\n<div data-type=\"title\">Properties of the Graph of \\(y={\\text{log}}_{a}x\\)<\/div>\n<table id=\"fs-id1167831882193\" class=\"unnumbered\" summary=\"Table has four columns. It shows that when a is greater than 1, the domain is 0 to infinity, the range is negative infinity to infinity, the x-intercept is 1, 0, there is no y intercept, the function contains a, 1 and , 1 over a, negative 1, the asymptote is the y axis 0, and the basic shape is increasing. It shows that when a is greater than 0 and less than 1, the domain is 0 to infinity, the range is negative infinity to infinity, the x-intercept is 1, 0, there is no y intercept, the function contains a, 1 and 1 over a, negative 1, the asymptote is the y axis, and the basic shape is decreasing.\" data-label=\"\">\n<thead>\n<tr>\n<th colspan=\"2\" data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">when<\/strong>\\(a&gt;1\\)<\/th>\n<th colspan=\"2\" data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">when<\/strong>\\(0&lt;a&lt;1\\)<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Domain<\/td>\n<td data-valign=\"top\" data-align=\"left\">\\(\\left(0,\\infty \\right)\\)<\/td>\n<td data-valign=\"top\" data-align=\"left\">Domain<\/td>\n<td data-valign=\"top\" data-align=\"left\">\\(\\left(0,\\infty \\right)\\)<\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Range<\/td>\n<td data-valign=\"top\" data-align=\"left\">\\(\\left(\\text{\u2212}\\infty ,\\infty \\right)\\)<\/td>\n<td data-valign=\"top\" data-align=\"left\">Range<\/td>\n<td data-valign=\"top\" data-align=\"left\">\\(\\left(\\text{\u2212}\\infty ,\\infty \\right)\\)<\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">\\(x\\)-intercept<\/td>\n<td data-valign=\"top\" data-align=\"left\">\\(\\left(1,0\\right)\\)<\/td>\n<td data-valign=\"top\" data-align=\"left\">\\(x\\)-intercept<\/td>\n<td data-valign=\"top\" data-align=\"left\">\\(\\left(1,0\\right)\\)<\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">\\(y\\)-intercept<\/td>\n<td data-valign=\"top\" data-align=\"left\">none<\/td>\n<td data-valign=\"top\" data-align=\"left\">\\(y\\)-intercept<\/td>\n<td data-valign=\"top\" data-align=\"left\">None<\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Contains<\/td>\n<td data-valign=\"top\" data-align=\"left\">\\(\\left(a,1\\right),\\)\\(\\left(\\frac{1}{a},-1\\right)\\)<\/td>\n<td data-valign=\"top\" data-align=\"left\">Contains<\/td>\n<td data-valign=\"top\" data-align=\"left\">\\(\\left(a,1\\right),\\)\\(\\left(\\frac{1}{a},-1\\right)\\)<\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Asymptote<\/td>\n<td data-valign=\"top\" data-align=\"left\">\\(y\\)-axis<\/td>\n<td data-valign=\"top\" data-align=\"left\">Asymptote<\/td>\n<td data-valign=\"top\" data-align=\"left\">\\(y\\)-axis<\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Basic shape<\/td>\n<td data-valign=\"top\" data-align=\"left\">increasing<\/td>\n<td data-valign=\"top\" data-align=\"left\">Basic shape<\/td>\n<td data-valign=\"top\" data-align=\"left\">Decreasing<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<span data-type=\"media\" data-alt=\"This figure shows that, for a greater than 1, the logarithmic curve going through the points (1 over a, negative 1), (1, 0), and (a, 1). This figure shows that, for a greater than 0 and less than 1, the logarithmic curve going through the points (a, 1), (1, 0), and (1 over a, negative 1).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/jkdcbcintermediatealgebra\/wp-content\/uploads\/sites\/1021\/2020\/05\/CNX_IntAlg_Figure_10_03_007_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows that, for a greater than 1, the logarithmic curve going through the points (1 over a, negative 1), (1, 0), and (a, 1). This figure shows that, for a greater than 0 and less than 1, the logarithmic curve going through the points (a, 1), (1, 0), and (1 over a, negative 1).\"><\/span>\n\n<\/div>\n<p id=\"fs-id1167835510177\">We talked earlier about how the logarithmic function \\({f}^{-1}\\left(x\\right)={\\text{log}}_{a}x\\) is the inverse of the exponential function \\(f\\left(x\\right)={a}^{x}.\\) The graphs in <a href=\"#CNX_IntAlg_Figure_10_03_008_img\" class=\"autogenerated-content\">(Figure)<\/a> show both the exponential (blue) and logarithmic (red) functions on the same graph for both \\(a&gt;1\\) and \\(0&lt;a&lt;1.\\)<\/p>\n\n<div class=\"bc-figure figure\" id=\"CNX_IntAlg_Figure_10_03_008_img\"><span data-type=\"media\" id=\"fs-id1167832074212\" data-alt=\"This figure shows that, for a greater than 1, the logarithmic curve going through the points (1 over a, negative 1), (1, 0), and (a, 1). It also shows the exponential curve going through the points (1, 1 over a), (0, 1), and (1, a) along with the line y equals x. The logarithmic curve is a mirror image of the exponential curve across the y equals x line. This figure shows that, for a greater than 0 and less than 1, the logarithmic curve going through the points (a, 1), (1, 0), and (1 over a, negative 1). It also shows the exponential curve going through the points (negative 1, 1 over a), (0, 1), and (1, a) along with the line y equals x. The logarithmic curve is a mirror image of the exponential curve across the y equals x line.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/jkdcbcintermediatealgebra\/wp-content\/uploads\/sites\/1021\/2020\/05\/CNX_IntAlg_Figure_10_03_008_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows that, for a greater than 1, the logarithmic curve going through the points (1 over a, negative 1), (1, 0), and (a, 1). It also shows the exponential curve going through the points (1, 1 over a), (0, 1), and (1, a) along with the line y equals x. The logarithmic curve is a mirror image of the exponential curve across the y equals x line. This figure shows that, for a greater than 0 and less than 1, the logarithmic curve going through the points (a, 1), (1, 0), and (1 over a, negative 1). It also shows the exponential curve going through the points (negative 1, 1 over a), (0, 1), and (1, a) along with the line y equals x. The logarithmic curve is a mirror image of the exponential curve across the y equals x line.\"><\/span><\/div>\n<p id=\"fs-id1167835417624\">Notice how the graphs are reflections of each other through the line \\(y=x.\\) We know this is true of inverse functions. Keeping a visual in your mind of these graphs will help you remember the domain and range of each function. Notice the <em data-effect=\"italics\">x<\/em>-axis is the horizontal asymptote for the exponential functions and the <em data-effect=\"italics\">y<\/em>-axis is the vertical asymptote for the logarithmic functions.<\/p>\n\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167834473423\">\n<h3 data-type=\"title\">Solve Logarithmic Equations<\/h3>\n<p id=\"fs-id1167832015982\">When we talked about exponential functions, we introduced the number <em data-effect=\"italics\">e<\/em>. Just as <em data-effect=\"italics\">e<\/em> was a base for an exponential function, it can be used a base for logarithmic functions too. The logarithmic function with base <em data-effect=\"italics\">e<\/em> is called the <span data-type=\"term\">natural logarithmic function<\/span>. The function \\(f\\left(x\\right)={\\text{log}}_{e}x\\) is generally written \\(f\\left(x\\right)=\\text{ln}\\phantom{\\rule{0.2em}{0ex}}x\\) and we read it as \u201cel en of \\(x.\u201d\\)<\/p>\n\n<div data-type=\"note\" id=\"fs-id1167835351566\">\n<div data-type=\"title\">Natural Logarithmic Function<\/div>\n<p id=\"fs-id1167835234222\">The function \\(f\\left(x\\right)=\\text{ln}\\phantom{\\rule{0.2em}{0ex}}x\\) is the <strong data-effect=\"bold\">natural logarithmic function<\/strong> with base \\(e,\\) where \\(x&gt;0.\\)<\/p>\n\n<div data-type=\"equation\" id=\"fs-id1167835318932\" class=\"unnumbered\" data-label=\"\">\\(y=\\text{ln}\\phantom{\\rule{0.2em}{0ex}}x\\phantom{\\rule{0.2em}{0ex}}\\text{is equivalent to}\\phantom{\\rule{0.2em}{0ex}}x={e}^{y}\\)<\/div>\n<\/div>\n<p id=\"fs-id1167834448872\">When the base of the logarithm function is 10, we call it the <span data-type=\"term\">common logarithmic function<\/span> and the base is not shown. If the base <em data-effect=\"italics\">a<\/em> of a logarithm is not shown, we assume it is 10.<\/p>\n\n<div data-type=\"note\" id=\"fs-id1167831214279\">\n<div data-type=\"title\">Common Logarithmic Function<\/div>\n<p id=\"fs-id1167835311021\">The function \\(f\\left(x\\right)=\\text{log}\\phantom{\\rule{0.2em}{0ex}}x\\) is the <strong data-effect=\"bold\">common logarithmic function<\/strong> with base\\(10\\), where \\(x&gt;0.\\)<\/p>\n\n<div data-type=\"equation\" id=\"fs-id1167831227947\" class=\"unnumbered\" data-label=\"\">\\(y=\\text{log}\\phantom{\\rule{0.2em}{0ex}}x\\phantom{\\rule{0.2em}{0ex}}\\text{is equivalent to}\\phantom{\\rule{0.2em}{0ex}}x={10}^{y}\\)<\/div>\n<\/div>\n<span data-type=\"media\" id=\"fs-id1167826967294\" data-alt=\"It will be important for you to use your calculator to evaluate both common and natural logarithms. Find the log and ln keys on your calculator.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/jkdcbcintermediatealgebra\/wp-content\/uploads\/sites\/1021\/2020\/05\/CNX_IntAlg_Figure_10_03_009_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"It will be important for you to use your calculator to evaluate both common and natural logarithms. Find the log and ln keys on your calculator.\"><\/span>\n<p id=\"fs-id1167835416834\">To solve logarithmic equations, one strategy is to change the equation to exponential form and then solve the exponential equation as we did before. As we solve logarithmic equations, \\(y={\\text{log}}_{a}x\\), we need to remember that for the base <em data-effect=\"italics\">a<\/em>, \\(a&gt;0\\) and \\(a\\ne 1.\\) Also, the domain is \\(x&gt;0.\\) Just as with radical equations, we must check our solutions to eliminate any extraneous solutions.<\/p>\n\n<div data-type=\"example\" id=\"fs-id1167826996479\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167834473642\">\n<div data-type=\"problem\" id=\"fs-id1167835311985\">\n<p id=\"fs-id1167835380844\">Solve: <span class=\"token\">\u24d0<\/span> \\({\\text{log}}_{a}49=2\\) and <span class=\"token\">\u24d1<\/span> \\(\\text{ln}\\phantom{\\rule{0.2em}{0ex}}x=3.\\)<\/p>\n\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835181600\">\n<p id=\"fs-id1167835240986\"><span class=\"token\">\u24d0<\/span><\/p>\n\n<div data-type=\"newline\"><\/div>\n\\(\\begin{array}{c}\\begin{array}{cccccc}&amp; &amp; &amp; \\hfill {\\mathrm{log}}_{a}49&amp; =\\hfill &amp; 2\\hfill \\\\ \\text{Rewrite in exponential form.}\\hfill &amp; &amp; &amp; \\hfill {a}^{2}&amp; =\\hfill &amp; 49\\hfill \\\\ \\text{Solve the equation using the square root property.}\\hfill &amp; &amp; &amp; \\hfill a&amp; =\\hfill &amp; \u00b17\\hfill \\end{array}\\phantom{\\rule{0.3em}{0ex}}\\hfill \\\\ \\begin{array}{c}\\text{The base cannot be negative, so we eliminate}\\hfill \\\\ a=-7.\\phantom{\\rule{18.5em}{0ex}}a=7,\\phantom{\\rule{0.5em}{0ex}}\\overline{)a=-7}\\hfill \\end{array}\\hfill \\\\ \\text{Check.}\\hfill \\\\ \\begin{array}{cccccccc}a=7\\hfill &amp; &amp; &amp; &amp; &amp; \\hfill {\\mathrm{log}}_{a}49&amp; =\\hfill &amp; 2\\hfill \\\\ &amp; &amp; &amp; &amp; &amp; \\hfill {\\mathrm{log}}_{7}49&amp; \\stackrel{?}{=}\\hfill &amp; 2\\hfill \\\\ &amp; &amp; &amp; &amp; &amp; \\hfill {7}^{2}&amp; \\stackrel{?}{=}\\hfill &amp; 49\\hfill \\\\ &amp; &amp; &amp; &amp; &amp; \\hfill 49&amp; =\\hfill &amp; 49\u2713\\hfill \\end{array}\\hfill &amp; &amp; \\end{array}\\)\n<p id=\"fs-id1167835259254\"><span class=\"token\">\u24d1<\/span><\/p>\n\n<div data-type=\"newline\"><\/div>\n\\(\\begin{array}{cccccc}&amp; &amp; &amp; \\hfill \\phantom{\\rule{8em}{0ex}}\\text{ln}\\phantom{\\rule{0.2em}{0ex}}x&amp; =\\hfill &amp; 3\\hfill \\\\ \\text{Rewrite in exponential form.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{7.7em}{0ex}}{e}^{3}&amp; =\\hfill &amp; x\\hfill \\\\ \\text{Check.}\\hfill &amp; &amp; &amp; &amp; &amp; \\\\ \\begin{array}{cccccccc}x={e}^{3}\\hfill &amp; &amp; &amp; &amp; &amp; \\hfill \\text{ln}\\phantom{\\rule{0.2em}{0ex}}x&amp; =\\hfill &amp; 3\\hfill \\\\ &amp; &amp; &amp; &amp; &amp; \\hfill \\text{ln}\\phantom{\\rule{0.2em}{0ex}}{e}^{3}&amp; \\stackrel{?}{=}\\hfill &amp; 3\\hfill \\\\ &amp; &amp; &amp; &amp; &amp; \\hfill {e}^{3}&amp; =\\hfill &amp; {e}^{3}\u2713\\hfill \\end{array}\\hfill \\end{array}\\)\n\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167826997664\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835253878\">\n<div data-type=\"problem\" id=\"fs-id1167835595145\">\n<p id=\"fs-id1167834134740\">Solve: <span class=\"token\">\u24d0<\/span> \\({\\text{log}}_{a}121=2\\) <span class=\"token\">\u24d1<\/span> \\(\\text{ln}\\phantom{\\rule{0.2em}{0ex}}x=7\\)<\/p>\n\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835511251\">\n<div data-type=\"newline\"><\/div>\n<span class=\"token\">\u24d0<\/span>\n<div data-type=\"newline\"><\/div>\n\\(a=11\\)\n<div data-type=\"newline\"><\/div>\n<span class=\"token\">\u24d1<\/span>\\(x={e}^{7}\\)\n\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167834432020\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835236467\">\n<div data-type=\"problem\" id=\"fs-id1167835330555\">\n<p id=\"fs-id1167826781244\">Solve: <span class=\"token\">\u24d0<\/span> \\({\\text{log}}_{a}64=3\\) <span class=\"token\">\u24d1<\/span> \\(\\text{ln}\\phantom{\\rule{0.2em}{0ex}}x=9\\)<\/p>\n\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831888043\">\n<div data-type=\"newline\"><\/div>\n<span class=\"token\">\u24d0<\/span>\n<div data-type=\"newline\"><\/div>\n\\(a=4\\)\n<div data-type=\"newline\"><\/div>\n<span class=\"token\">\u24d1<\/span>\\(x={e}^{9}\\)\n\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"example\" id=\"fs-id1167831040648\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167835257318\">\n<div data-type=\"problem\" id=\"fs-id1167830697885\">\n<p id=\"fs-id1167831908789\">Solve: <span class=\"token\">\u24d0<\/span> \\({\\text{log}}_{2}\\left(3x-5\\right)=4\\) and <span class=\"token\">\u24d1<\/span> \\(\\text{ln}\\phantom{\\rule{0.2em}{0ex}}{e}^{2x}=4.\\)<\/p>\n\n<\/div>\n<div data-type=\"solution\">\n<p id=\"fs-id1167835173443\"><span class=\"token\">\u24d0<\/span><\/p>\n\n<div data-type=\"newline\"><\/div>\n\\(\\begin{array}{cccccc}&amp; &amp; &amp; \\hfill {\\mathrm{log}}_{2}\\left(3x-5\\right)&amp; =\\hfill &amp; 4\\hfill \\\\ \\text{Rewrite in exponential form.}\\hfill &amp; &amp; &amp; \\hfill {2}^{4}&amp; =\\hfill &amp; 3x-5\\hfill \\\\ \\text{Simplify.}\\hfill &amp; &amp; &amp; \\hfill 16&amp; =\\hfill &amp; 3x-5\\hfill \\\\ \\text{Solve the equation.}\\hfill &amp; &amp; &amp; \\hfill 21&amp; =\\hfill &amp; 3x\\hfill \\\\ &amp; &amp; &amp; \\hfill 7&amp; =\\hfill &amp; x\\hfill \\\\ \\text{Check.}\\hfill &amp; &amp; &amp; &amp; &amp; \\\\ \\begin{array}{cccccccc}x=7\\hfill &amp; &amp; &amp; &amp; &amp; \\hfill {\\mathrm{log}}_{2}\\left(3x-5\\right)&amp; =\\hfill &amp; 4\\hfill \\\\ &amp; &amp; &amp; &amp; &amp; \\hfill {\\mathrm{log}}_{2}\\left(3\\cdot 7-5\\right)&amp; \\stackrel{?}{=}\\hfill &amp; 4\\hfill \\\\ &amp; &amp; &amp; &amp; &amp; \\hfill {\\mathrm{log}}_{2}\\left(16\\right)&amp; \\stackrel{?}{=}\\hfill &amp; 4\\hfill \\\\ &amp; &amp; &amp; &amp; &amp; \\hfill {2}^{4}&amp; \\stackrel{?}{=}\\hfill &amp; 16\\hfill \\\\ &amp; &amp; &amp; &amp; &amp; \\hfill 16&amp; =\\hfill &amp; 16\u2713\\hfill \\end{array}\\hfill \\end{array}\\)\n<p id=\"fs-id1167831922589\"><span class=\"token\">\u24d1<\/span><\/p>\n\n<div data-type=\"newline\"><\/div>\n\\(\\begin{array}{cccccc}&amp; &amp; &amp; \\hfill \\mathrm{ln}\\phantom{\\rule{0.2em}{0ex}}{e}^{2x}&amp; =\\hfill &amp; 4\\hfill \\\\ \\text{Rewrite in exponential form.}\\hfill &amp; &amp; &amp; \\hfill {e}^{4}&amp; =\\hfill &amp; {e}^{2x}\\hfill \\\\ \\text{Since the bases are the same the exponents are equal.}\\hfill &amp; &amp; &amp; \\hfill 4&amp; =\\hfill &amp; 2x\\hfill \\\\ \\text{Solve the equation.}\\hfill &amp; &amp; &amp; \\hfill 2&amp; =\\hfill &amp; x\\hfill \\\\ \\text{Check.}\\hfill &amp; &amp; &amp; &amp; &amp; \\\\ \\begin{array}{cccccccc}x=2\\hfill &amp; &amp; &amp; &amp; &amp; \\hfill \\mathrm{ln}\\phantom{\\rule{0.2em}{0ex}}{e}^{2x}&amp; =\\hfill &amp; 4\\hfill \\\\ &amp; &amp; &amp; &amp; &amp; \\hfill \\mathrm{ln}\\phantom{\\rule{0.2em}{0ex}}{e}^{2\u00b72}&amp; \\stackrel{?}{=}\\hfill &amp; 4\\hfill \\\\ &amp; &amp; &amp; &amp; &amp; \\hfill \\mathrm{ln}\\phantom{\\rule{0.2em}{0ex}}{e}^{4}&amp; \\stackrel{?}{=}\\hfill &amp; 4\\hfill \\\\ &amp; &amp; &amp; &amp; &amp; \\hfill {e}^{4}&amp; =\\hfill &amp; {e}^{4}\u2713\\hfill \\end{array}\\hfill \\end{array}\\)\n\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167834423068\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167834423072\">\n<div data-type=\"problem\" id=\"fs-id1167834556830\">\n<p id=\"fs-id1167834556832\">Solve: <span class=\"token\">\u24d0<\/span> \\({\\text{log}}_{2}\\left(5x-1\\right)=6\\) <span class=\"token\">\u24d1<\/span> \\(\\text{ln}\\phantom{\\rule{0.2em}{0ex}}{e}^{3x}=6\\)<\/p>\n\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834462946\">\n<div data-type=\"newline\"><\/div>\n<span class=\"token\">\u24d0<\/span>\n<div data-type=\"newline\"><\/div>\n\\(x=13\\)\n<div data-type=\"newline\"><\/div>\n<span class=\"token\">\u24d1<\/span>\\(x=2\\)\n\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835421044\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835421048\">\n<div data-type=\"problem\" id=\"fs-id1167835342572\">\n<p id=\"fs-id1167835342574\">Solve: <span class=\"token\">\u24d0<\/span> \\({\\text{log}}_{3}\\left(4x+3\\right)=3\\) <span class=\"token\">\u24d1<\/span> \\(\\text{ln}\\phantom{\\rule{0.2em}{0ex}}{e}^{4x}=4\\)<\/p>\n\n<\/div>\n<div data-type=\"solution\">\n<div data-type=\"newline\"><\/div>\n<span class=\"token\">\u24d0<\/span>\n<div data-type=\"newline\"><\/div>\n\\(x=6\\)\n<div data-type=\"newline\"><\/div>\n<span class=\"token\">\u24d1<\/span>\\(x=1\\)\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167831920506\">\n<h3 data-type=\"title\">Use Logarithmic Models in Applications<\/h3>\n<p id=\"fs-id1167834301111\">There are many applications that are modeled by logarithmic equations. We will first look at the logarithmic equation that gives the decibel (dB) level of sound. Decibels range from 0, which is barely audible to 160, which can rupture an eardrum. The \\({10}^{-12}\\) in the formula represents the intensity of sound that is barely audible.<\/p>\n\n<div data-type=\"note\" id=\"fs-id1167827987443\">\n<div data-type=\"title\">Decibel Level of Sound<\/div>\n<p id=\"fs-id1167827987448\">The loudness level, <em data-effect=\"italics\">D<\/em>, measured in decibels, of a sound of intensity, <em data-effect=\"italics\">I<\/em>, measured in watts per square inch is<\/p>\n\n<div data-type=\"equation\" id=\"fs-id1167834191577\" class=\"unnumbered\" data-label=\"\">\\(D=10\\phantom{\\rule{0.2em}{0ex}}\\text{log}\\left(\\frac{I}{{10}^{-12}}\\right)\\)<\/div>\n<\/div>\n<div data-type=\"example\" id=\"fs-id1167834062530\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167834062533\">\n<div data-type=\"problem\" id=\"fs-id1167834062535\">\n<p id=\"fs-id1167832149826\">Extended exposure to noise that measures 85 dB can cause permanent damage to the inner ear which will result in hearing loss. What is the decibel level of music coming through ear phones with intensity \\({10}^{-2}\\) watts per square inch?<\/p>\n\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834098991\">\n<table id=\"fs-id1167835359596\" class=\"unnumbered unstyled\" summary=\"We start with D equals 10 times log of the quantity I over 10 to the negative 12 power. We substitute in the intensity level I to obtain D equals 10 log of the quantity10 to the negative 2 power over 10 to the negative 12 power. We then simplify to obtain D equals 10 times log of 10 to the 10 power. Since log of 10 to the 10 equals 10, we have that D equals 10 times 10. Multiplying gives that D equals 100. Hence, the decibel level of music coming through earphones is 100 dB.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834183461\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/jkdcbcintermediatealgebra\/wp-content\/uploads\/sites\/1021\/2020\/05\/CNX_IntAlg_Figure_10_03_010a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Substitute in the intensity level, <em data-effect=\"italics\">I.<\/em><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167830961870\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/jkdcbcintermediatealgebra\/wp-content\/uploads\/sites\/1021\/2020\/05\/CNX_IntAlg_Figure_10_03_010b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832066044\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/jkdcbcintermediatealgebra\/wp-content\/uploads\/sites\/1021\/2020\/05\/CNX_IntAlg_Figure_10_03_010c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Since \\(\\text{log}{10}^{10}=10.\\)<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834094687\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/jkdcbcintermediatealgebra\/wp-content\/uploads\/sites\/1021\/2020\/05\/CNX_IntAlg_Figure_10_03_010d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Multiply.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835334481\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/jkdcbcintermediatealgebra\/wp-content\/uploads\/sites\/1021\/2020\/05\/CNX_IntAlg_Figure_10_03_010e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"left\">The decibel level of music coming through earphones is 100 dB.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167831106999\" class=\"try\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167835596124\">\n<p id=\"fs-id1167835329663\">What is the decibel level of one of the new quiet dishwashers with intensity \\({10}^{-7}\\) watts per square inch?<\/p>\n\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834532484\">\n<p id=\"fs-id1167834532486\">The quiet dishwashers have a decibel level of 50 dB.<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167826778780\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167832067744\">\n<div data-type=\"problem\" id=\"fs-id1167832067746\">\n<p id=\"fs-id1167832067748\">What is the decibel level heavy city traffic with intensity \\({10}^{-3}\\) watts per square inch?<\/p>\n\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835377416\">\n<p id=\"fs-id1167835377418\">The decibel level of heavy traffic is 90 dB.<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167835595015\">The magnitude \\(R\\) of an earthquake is measured by a logarithmic scale called the Richter scale. The model is \\(R=\\text{log}\\phantom{\\rule{0.2em}{0ex}}I,\\) where \\(I\\) is the intensity of the shock wave. This model provides a way to measure <span data-type=\"term\" class=\"no-emphasis\">earthquake intensity<\/span>.<\/p>\n\n<div data-type=\"note\" id=\"fs-id1167826857422\">\n<div data-type=\"title\">Earthquake Intensity<\/div>\n<p id=\"fs-id1167835235048\">The magnitude <em data-effect=\"italics\">R<\/em> of an earthquake is measured by \\(R=\\text{log}\\phantom{\\rule{0.2em}{0ex}}I,\\) where <em data-effect=\"italics\">I<\/em> is the intensity of its shock wave.<\/p>\n\n<\/div>\n<div data-type=\"example\" id=\"fs-id1167831893196\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167834537686\">\n<div data-type=\"problem\" id=\"fs-id1167834537688\">\n<p id=\"fs-id1167834537690\">In 1906, San Francisco experienced an intense earthquake with a magnitude of 7.8 on the Richter scale. Over 80% of the city was destroyed by the resulting fires. In 2014, Los Angeles experienced a moderate earthquake that measured 5.1 on the Richter scale and caused ?108 million dollars of damage. Compare the intensities of the two earthquakes.<\/p>\n\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167832066561\">\n<p id=\"fs-id1167832066563\">To compare the intensities, we first need to convert the magnitudes to intensities using the log formula. Then we will set up a ratio to compare the intensities.<\/p>\n<p id=\"fs-id1167835353005\">\\(\\begin{array}{cccc}\\text{Convert the magnitudes to intensities.}\\hfill &amp; &amp; &amp; \\phantom{\\rule{0.6em}{0ex}}R=\\text{log}\\phantom{\\rule{0.2em}{0ex}}I\\hfill \\\\ \\phantom{\\rule{2em}{0ex}}\\text{1906 earthquake}\\hfill &amp; &amp; &amp; 7.8=\\text{log}\\phantom{\\rule{0.2em}{0ex}}I\\hfill \\\\ \\phantom{\\rule{2em}{0ex}}\\text{Convert to exponential form.}\\hfill &amp; &amp; &amp; \\phantom{\\rule{0.86em}{0ex}}I={10}^{7.8}\\hfill \\\\ \\\\ \\\\ \\phantom{\\rule{2em}{0ex}}\\text{2014 earthquake}\\hfill &amp; &amp; &amp; 5.1=\\text{log}\\phantom{\\rule{0.2em}{0ex}}I\\hfill \\\\ \\phantom{\\rule{2em}{0ex}}\\text{Convert to exponential form.}\\hfill &amp; &amp; &amp; \\phantom{\\rule{0.92em}{0ex}}I={10}^{5.1}\\hfill \\\\ \\text{Form a ratio of the intensities.}\\hfill &amp; &amp; &amp; \\frac{\\text{Intensity}\\phantom{\\rule{0.2em}{0ex}}\\text{for}\\phantom{\\rule{0.2em}{0ex}}1906}{\\text{Intensity}\\phantom{\\rule{0.2em}{0ex}}\\text{for}\\phantom{\\rule{0.2em}{0ex}}2014}\\hfill \\\\ \\text{Substitute in the values.}\\hfill &amp; &amp; &amp; \\phantom{\\rule{1.5em}{0ex}}\\frac{{10}^{7.8}}{{10}^{5.1}}\\hfill \\\\ \\text{Divide by subtracting the exponents.}\\hfill &amp; &amp; &amp; \\phantom{\\rule{1.5em}{0ex}}{10}^{2.7}\\hfill \\\\ \\text{Evaluate.}\\hfill &amp; &amp; &amp; \\phantom{\\rule{1.5em}{0ex}}501\\hfill \\\\ &amp; &amp; &amp; \\begin{array}{c}\\text{The intensity of the 1906 earthquake}\\hfill \\\\ \\text{was about 501 times the intensity of}\\hfill \\\\ \\text{the 2014 earthquake.}\\hfill \\end{array}\\hfill \\end{array}\\)<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167827943038\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167827943041\">\n<div data-type=\"problem\" id=\"fs-id1167831826460\">\n<p id=\"fs-id1167831826462\">In 1906, San Francisco experienced an intense earthquake with a magnitude of 7.8 on the Richter scale. In 1989, the Loma Prieta earthquake also affected the San Francisco area, and measured 6.9 on the Richter scale. Compare the intensities of the two earthquakes.<\/p>\n\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835229713\">\n<p id=\"fs-id1167835229715\">The intensity of the 1906 earthquake was about 8 times the intensity of the 1989 earthquake.<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835345396\" class=\"try\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167835217831\">\n<p id=\"fs-id1167835217833\">In 2014, Chile experienced an intense earthquake with a magnitude of 8.2 on the Richter scale. In 2014, Los Angeles also experienced an earthquake which measured 5.1 on the Richter scale. Compare the intensities of the two earthquakes.<\/p>\n\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834131115\">\n<p id=\"fs-id1167831922298\">The intensity of the earthquake in Chile was about 1,259 times the intensity of the earthquake in Los Angeles.<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167834065272\" class=\"media-2\">\n<p id=\"fs-id1167834065275\">Access these online resources for additional instruction and practice with evaluating and graphing logarithmic functions.<\/p>\n\n<ul id=\"fs-id1167835329751\" data-display=\"block\">\n \t<li><a href=\"https:\/\/openstax.org\/l\/37logasexponent\">Re-writing logarithmic equations in exponential form<\/a><\/li>\n \t<li><a href=\"https:\/\/openstax.org\/l\/37Simplifylog\">Simplifying Logarithmic Expressions<\/a><\/li>\n \t<li><a href=\"https:\/\/openstax.org\/l\/37Graphlog\">Graphing logarithmic functions<\/a><\/li>\n \t<li><a href=\"https:\/\/openstax.org\/l\/37Finddecibel\">Using logarithms to calculate decibel levels<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167832076686\">\n<h3 data-type=\"title\">Key Concepts<\/h3>\n<ul id=\"fs-id1167826827794\" data-bullet-style=\"bullet\">\n \t<li><strong data-effect=\"bold\">Properties of the Graph of<\/strong>\\(y={\\text{log}}_{a}x:\\)\n<div data-type=\"newline\"><\/div>\n<table id=\"fs-id1167835329636\" class=\"unnumbered\" summary=\"Table has four columns. It shows that when a is greater than 1, the domain is 0 to infinity, the range is negative infinity to infinity, the x-intercept is 1, 0, there is no y intercept, the function contains a, 1 and , 1 over a, negative 1, the asymptote is the y axis 0, and the basic shape is increasing. It shows that when a is greater than 0 and less than 1, the domain is 0 to infinity, the range is negative infinity to infinity, the x-intercept is 1, 0, there is no y intercept, the function contains a, 1 and 1 over a, negative 1, the asymptote is the y axis, and the basic shape is decreasing.\" data-label=\"\">\n<thead>\n<tr>\n<th colspan=\"2\" data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">when<\/strong>\\(a&gt;1\\)<\/th>\n<th colspan=\"2\" data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">when<\/strong>\\(0&lt;a&lt;1\\)<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Domain<\/td>\n<td data-valign=\"top\" data-align=\"left\">\\(\\left(0,\\infty \\right)\\)<\/td>\n<td data-valign=\"top\" data-align=\"left\">Domain<\/td>\n<td data-valign=\"top\" data-align=\"left\">\\(\\left(0,\\infty \\right)\\)<\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Range<\/td>\n<td data-valign=\"top\" data-align=\"left\">\\(\\left(\\text{\u2212}\\infty ,\\infty \\right)\\)<\/td>\n<td data-valign=\"top\" data-align=\"left\">Range<\/td>\n<td data-valign=\"top\" data-align=\"left\">\\(\\left(\\text{\u2212}\\infty ,\\infty \\right)\\)<\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\"><em data-effect=\"italics\">x<\/em>-intercept<\/td>\n<td data-valign=\"top\" data-align=\"left\">\\(\\left(1,0\\right)\\)<\/td>\n<td data-valign=\"top\" data-align=\"left\"><em data-effect=\"italics\">x<\/em>-intercept<\/td>\n<td data-valign=\"top\" data-align=\"left\">\\(\\left(1,0\\right)\\)<\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\"><em data-effect=\"italics\">y<\/em>-intercept<\/td>\n<td data-valign=\"top\" data-align=\"left\">none<\/td>\n<td data-valign=\"top\" data-align=\"left\"><em data-effect=\"italics\">y<\/em>-intercept<\/td>\n<td data-valign=\"top\" data-align=\"left\">none<\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Contains<\/td>\n<td data-valign=\"top\" data-align=\"left\">\\(\\left(a,1\\right),\\)\\(\\left(\\frac{1}{a},-1\\right)\\)<\/td>\n<td data-valign=\"top\" data-align=\"left\">Contains<\/td>\n<td data-valign=\"top\" data-align=\"left\">\\(\\left(a,1\\right),\\)\\(\\left(\\frac{1}{a},-1\\right)\\)<\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Asymptote<\/td>\n<td data-valign=\"top\" data-align=\"left\"><em data-effect=\"italics\">y<\/em>-axis<\/td>\n<td data-valign=\"top\" data-align=\"left\">Asymptote<\/td>\n<td data-valign=\"top\" data-align=\"left\"><em data-effect=\"italics\">y<\/em>-axis<\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Basic shape<\/td>\n<td data-valign=\"top\" data-align=\"left\">increasing<\/td>\n<td data-valign=\"top\" data-align=\"left\">Basic shape<\/td>\n<td data-valign=\"top\" data-align=\"left\">decreasing<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div data-type=\"newline\"><\/div>\n<span data-type=\"media\" id=\"fs-id1167835380217\" data-alt=\"This figure shows that, for a greater than 1, the logarithmic curve going through the points (1 over a, negative 1), (1, 0), and (a, 1). This figure shows that, for a greater than 0 and less than 1, the logarithmic curve going through the points (a, 1), (1, 0), and (1 over a, negative 1).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/jkdcbcintermediatealgebra\/wp-content\/uploads\/sites\/1021\/2020\/05\/CNX_IntAlg_Figure_10_03_012_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows that, for a greater than 1, the logarithmic curve going through the points (1 over a, negative 1), (1, 0), and (a, 1). This figure shows that, for a greater than 0 and less than 1, the logarithmic curve going through the points (a, 1), (1, 0), and (1 over a, negative 1).\"><\/span><\/li>\n \t<li><strong data-effect=\"bold\">Decibel Level of Sound:<\/strong> The loudness level, \\(D\\), measured in decibels, of a sound of intensity, \\(I\\), measured in watts per square inch is \\(D=10\\text{log}\\left(\\frac{I}{{10}^{-12}}\\right).\\)<\/li>\n \t<li><strong data-effect=\"bold\">Earthquake Intensity:<\/strong> The magnitude \\(R\\) of an earthquake is measured by \\(R=\\text{log}\\phantom{\\rule{0.2em}{0ex}}I,\\) where \\(I\\) is the intensity of its shock wave.<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167826874229\">\n<div class=\"practice-perfect\" data-depth=\"2\">\n<h4 data-type=\"title\">Practice Makes Perfect<\/h4>\n<p id=\"fs-id1167830757469\"><strong data-effect=\"bold\">Convert Between Exponential and Logarithmic Form<\/strong><\/p>\nIn the following exercises, convert from exponential to logarithmic form.\n<div data-type=\"exercise\" id=\"fs-id1167835610117\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835610119\">\n<p id=\"fs-id1167835610121\">\\({4}^{2}=16\\)<\/p>\n\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835511476\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835511478\">\n<p id=\"fs-id1167835614916\">\\({2}^{5}=32\\)<\/p>\n\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834402936\">\n<p id=\"fs-id1167834402938\">\\({\\text{log}}_{2}32=5\\)<\/p>\n\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835378536\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835351753\">\n<p id=\"fs-id1167835351755\">\\({3}^{3}=27\\)<\/p>\n\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826828342\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167826828344\">\n<p id=\"fs-id1167826828346\">\\({5}^{3}=125\\)<\/p>\n\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834064488\">\n<p id=\"fs-id1167834064490\">\\({\\text{log}}_{5}125=3\\)<\/p>\n\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167830698573\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167830698575\">\n<p id=\"fs-id1167835376182\">\\({10}^{3}=1000\\)<\/p>\n\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826781041\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167826781043\">\n<p id=\"fs-id1167826781046\">\\({10}^{-2}=\\frac{1}{100}\\)<\/p>\n\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834186366\">\n<p id=\"fs-id1167834186368\">\\(\\text{log}\\frac{1}{100}=-2\\)<\/p>\n\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831970028\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167831970030\">\n<p id=\"fs-id1167834534486\">\\({x}^{\\frac{1}{2}}=\\sqrt{3}\\)<\/p>\n\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831872216\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167831872218\">\n<p id=\"fs-id1167835511349\">\\({x}^{\\frac{1}{3}}=\\sqrt[3]{6}\\)<\/p>\n\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834429528\">\n<p id=\"fs-id1167834429530\">\\({\\text{log}}_{x}\\sqrt[3]{6}=\\frac{1}{3}\\)<\/p>\n\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832043157\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835167487\">\n<p id=\"fs-id1167835167489\">\\({32}^{x}=\\sqrt[4]{32}\\)<\/p>\n\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831081691\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834058866\">\n<p id=\"fs-id1167834058868\">\\({17}^{x}=\\sqrt[5]{17}\\)<\/p>\n\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835356149\">\n<p id=\"fs-id1167830914966\">\\({\\text{log}}_{17}\\sqrt[5]{17}=x\\)<\/p>\n\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835240982\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167826995298\">\n<p id=\"fs-id1167826995300\">\\({\\left(\\frac{1}{4}\\right)}^{2}=\\frac{1}{16}\\)<\/p>\n\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835340501\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834555158\">\n<p id=\"fs-id1167834555160\">\\({\\left(\\frac{1}{3}\\right)}^{4}=\\frac{1}{81}\\)<\/p>\n\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835283043\">\n<p id=\"fs-id1167835283045\">\\({\\text{log}}_{\\frac{1}{3}}\\frac{1}{81}=4\\)<\/p>\n\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167828411027\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835311339\">\n<p id=\"fs-id1167835311341\">\\({3}^{-2}=\\frac{1}{9}\\)<\/p>\n\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167830865339\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167830865341\">\n<p id=\"fs-id1167835511733\">\\({4}^{-3}=\\frac{1}{64}\\)<\/p>\n\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835305342\">\n<p id=\"fs-id1167835305344\">\\({\\text{log}}_{4}\\frac{1}{64}=-3\\)<\/p>\n\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826799078\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167826799080\">\n<p id=\"fs-id1167826799082\">\\({e}^{x}=6\\)<\/p>\n\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826997372\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834289504\">\n<p id=\"fs-id1167834289506\">\\({e}^{3}=x\\)<\/p>\n\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835374439\">\n<p id=\"fs-id1167834403039\">\\(\\text{ln}\\phantom{\\rule{0.2em}{0ex}}x=3\\)<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id1167831031378\">In the following exercises, convert each logarithmic equation to exponential form.<\/p>\n\n<div data-type=\"exercise\" id=\"fs-id1167831031381\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167831031383\">\n<p id=\"fs-id1167835332820\">\\(3={\\text{log}}_{4}64\\)<\/p>\n\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832152944\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834402694\">\n<p id=\"fs-id1167834402696\">\\(6={\\text{log}}_{2}64\\)<\/p>\n\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835302082\">\n<p id=\"fs-id1167835302084\">\\(64={2}^{6}\\)<\/p>\n\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834556107\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834556109\">\n<p id=\"fs-id1167834556111\">\\(4={\\text{log}}_{x}81\\)<\/p>\n\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835351507\" class=\"material-set-2\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167831076443\">\\(5={\\text{log}}_{x}32\\)<\/p>\n\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835511183\">\n<p id=\"fs-id1167835511185\">\\(32={x}^{5}\\)<\/p>\n\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835226504\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167832195745\">\n<p id=\"fs-id1167832195747\">\\(0={\\text{log}}_{12}1\\)<\/p>\n\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835510915\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835510918\">\n<p id=\"fs-id1167826994101\">\\(0={\\text{log}}_{7}1\\)<\/p>\n\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835331715\">\n<p id=\"fs-id1167835331717\">\\(1={7}^{0}\\)<\/p>\n\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835267509\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167826778870\">\n<p id=\"fs-id1167826778873\">\\(1={\\text{log}}_{3}3\\)<\/p>\n\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835336419\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835336421\">\n<p id=\"fs-id1167835336423\">\\(1={\\text{log}}_{9}9\\)<\/p>\n\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834246582\">\n<p id=\"fs-id1167835614859\">\\(9={9}^{1}\\)<\/p>\n\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167828447061\">\n<p id=\"fs-id1167831958004\">\\(-4={\\text{log}}_{10}\\frac{1}{10,000}\\)<\/p>\n\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832212044\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835243983\">\n<p id=\"fs-id1167835243985\">\\(3={\\text{log}}_{10}1,000\\)<\/p>\n\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834195105\">\n<p id=\"fs-id1167834195107\">\\(1,000={10}^{3}\\)<\/p>\n\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834300253\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834300255\">\n<p id=\"fs-id1167835373839\">\\(5={\\text{log}}_{e}x\\)<\/p>\n\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832058780\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167826983132\">\n<p id=\"fs-id1167826983134\">\\(x={\\text{log}}_{e}43\\)<\/p>\n\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835376396\">\n<p id=\"fs-id1167835376398\">\\(43={e}^{x}\\)<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id1167834058960\"><strong data-effect=\"bold\">Evaluate Logarithmic Functions<\/strong><\/p>\n<p id=\"fs-id1167830836813\">In the following exercises, find the value of \\(x\\) in each logarithmic equation.<\/p>\n\n<div data-type=\"exercise\" id=\"fs-id1167832123885\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167832123887\">\n<p id=\"fs-id1167832123889\">\\({\\text{log}}_{x}49=2\\)<\/p>\n\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834063090\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834063092\">\n<p id=\"fs-id1167834063094\">\\({\\text{log}}_{x}121=2\\)<\/p>\n\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834098316\">\n<p id=\"fs-id1167834098318\">\\(x=11\\)<\/p>\n\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832058679\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167831846682\">\n<p id=\"fs-id1167831846684\">\\({\\text{log}}_{x}27=3\\)<\/p>\n\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826799348\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167831923461\">\n<p id=\"fs-id1167831923463\">\\({\\text{log}}_{x}64=3\\)<\/p>\n\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834506100\">\n<p id=\"fs-id1167834506103\">\\(x=4\\)<\/p>\n\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835350317\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835350319\">\n<p id=\"fs-id1167834395147\">\\({\\text{log}}_{3}x=4\\)<\/p>\n\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835375306\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835375308\">\n<p id=\"fs-id1167835358824\">\\({\\text{log}}_{5}x=3\\)<\/p>\n\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834462750\">\n<p id=\"fs-id1167835365659\">\\(x=125\\)<\/p>\n\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835236219\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835236222\">\n<p id=\"fs-id1167835236224\">\\({\\text{log}}_{2}x=-6\\)<\/p>\n\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167830914861\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835468575\">\n<p id=\"fs-id1167835468577\">\\({\\text{log}}_{3}x=-5\\)<\/p>\n\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834246985\">\n<p id=\"fs-id1167834246987\">\\(x=\\frac{1}{243}\\)<\/p>\n\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835258321\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835258323\">\n<p id=\"fs-id1167835258325\">\\({\\text{log}}_{\\frac{1}{4}}\\frac{1}{16}=x\\)<\/p>\n\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835420083\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835420086\">\n<p id=\"fs-id1167835351531\">\\({\\text{log}}_{\\frac{1}{3}}\\frac{1}{9}=x\\)<\/p>\n\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834505061\">\n<p id=\"fs-id1167834505063\">\\(x=2\\)<\/p>\n\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831882568\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167831882570\">\n<p id=\"fs-id1167831066105\">\\({\\text{log}}_{\\frac{1}{4}}64=x\\)<\/p>\n\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831115372\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835339774\">\n<p id=\"fs-id1167835339776\">\\({\\text{log}}_{\\frac{1}{9}}81=x\\)<\/p>\n\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167830858557\">\n<p id=\"fs-id1167830858559\">\\(x=-2\\)<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id1167834252266\">In the following exercises, find the exact value of each logarithm without using a calculator.<\/p>\n\n<div data-type=\"exercise\" id=\"fs-id1167831881072\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167831881075\">\n<p id=\"fs-id1167831881077\">\\({\\text{log}}_{7}49\\)<\/p>\n\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834184863\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834184865\">\n<p id=\"fs-id1167834184868\">\\({\\text{log}}_{6}36\\)<\/p>\n\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835376014\">\n<p id=\"fs-id1167831823260\">2<\/p>\n\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835356037\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835356039\">\n<p id=\"fs-id1167835356042\">\\({\\text{log}}_{4}1\\)<\/p>\n\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835339780\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835339782\">\n<p id=\"fs-id1167835339784\">\\({\\text{log}}_{5}1\\)<\/p>\n\n<\/div>\n<div data-type=\"solution\">\n<p id=\"fs-id1167835339616\">0<\/p>\n\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835200134\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835200136\">\n<p id=\"fs-id1167835200139\">\\({\\text{log}}_{16}4\\)<\/p>\n\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834556258\" class=\"material-set-2\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167834556262\">\\({\\text{log}}_{27}3\\)<\/p>\n\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834228002\">\n<p id=\"fs-id1167834228004\">\\(\\frac{1}{3}\\)<\/p>\n\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826779062\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167826779064\">\n<p id=\"fs-id1167826779066\">\\({\\text{log}}_{\\frac{1}{2}}2\\)<\/p>\n\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834536087\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834536089\">\n<p id=\"fs-id1167834193412\">\\({\\text{log}}_{\\frac{1}{2}}4\\)<\/p>\n\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826938274\">\n<p id=\"fs-id1167826938277\">\\(-2\\)<\/p>\n\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835513008\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167831871914\">\n<p id=\"fs-id1167831871917\">\\({\\text{log}}_{2}\\frac{1}{16}\\)<\/p>\n\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832041687\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167832041689\">\n<p id=\"fs-id1167832041691\">\\({\\text{log}}_{3}\\frac{1}{27}\\)<\/p>\n\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835596456\">\n<p id=\"fs-id1167835596458\">\\(-3\\)<\/p>\n\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831969720\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167828421347\">\n<p id=\"fs-id1167828421349\">\\({\\text{log}}_{4}\\frac{1}{16}\\)<\/p>\n\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834195940\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834195942\">\n<p id=\"fs-id1167834195944\">\\({\\text{log}}_{9}\\frac{1}{81}\\)<\/p>\n\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826864570\">\n<p id=\"fs-id1167826864572\">\\(-2\\)<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id1167832126086\"><strong data-effect=\"bold\">Graph Logarithmic Functions<\/strong><\/p>\n<p id=\"fs-id1167835529367\">In the following exercises, graph each logarithmic function.<\/p>\n\n<div data-type=\"exercise\" id=\"fs-id1167835529370\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835257995\">\n<p id=\"fs-id1167835257997\">\\(y={\\text{log}}_{2}x\\)<\/p>\n\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835377701\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835377703\">\n<p id=\"fs-id1167832066676\">\\(y={\\text{log}}_{4}x\\)<\/p>\n\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835381928\"><span data-type=\"media\" id=\"fs-id1167835381931\" data-alt=\"This figure shows the logarithmic curve going through the points (1 over 4, negative 1), (1, 0), and (4, 1).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/jkdcbcintermediatealgebra\/wp-content\/uploads\/sites\/1021\/2020\/05\/CNX_IntAlg_Figure_10_03_306_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows the logarithmic curve going through the points (1 over 4, negative 1), (1, 0), and (4, 1).\"><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826802368\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167831880182\">\n<p id=\"fs-id1167831880184\">\\(y={\\text{log}}_{6}x\\)<\/p>\n\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826997598\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167826997601\">\n<p id=\"fs-id1167834324636\">\\(y={\\text{log}}_{7}x\\)<\/p>\n\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835317768\"><span data-type=\"media\" id=\"fs-id1167835317771\" data-alt=\"This figure shows that the logarithmic curve going through the points (1 over 7, negative 1), (1, 0), and (7, 1).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/jkdcbcintermediatealgebra\/wp-content\/uploads\/sites\/1021\/2020\/05\/CNX_IntAlg_Figure_10_03_308_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows that the logarithmic curve going through the points (1 over 7, negative 1), (1, 0), and (7, 1).\"><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831920466\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167831920468\">\n<p id=\"fs-id1167831920470\">\\(y={\\text{log}}_{1.5}x\\)<\/p>\n\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826864301\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167826864303\">\n<p id=\"fs-id1167826864305\">\\(y={\\text{log}}_{2.5}x\\)<\/p>\n\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835284763\"><span data-type=\"media\" id=\"fs-id1167831881934\" data-alt=\"This figure shows the logarithmic curve going through the points (2 over 5, negative 1), (1, 0), and (2.5, 1).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/jkdcbcintermediatealgebra\/wp-content\/uploads\/sites\/1021\/2020\/05\/CNX_IntAlg_Figure_10_03_310_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows the logarithmic curve going through the points (2 over 5, negative 1), (1, 0), and (2.5, 1).\"><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834120868\" class=\"material-set-2\">\n<div data-type=\"problem\">\n\n\\(y={\\text{log}}_{\\frac{1}{3}}x\\)\n\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835229539\">\n\n\\(y={\\text{log}}_{\\frac{1}{5}}x\\)\n\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835287695\"><span data-type=\"media\" id=\"fs-id1167835287698\" data-alt=\"This figure shows the logarithmic curve going through the points (1 over 5, 1), (1, 0), and (5, negative 1).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/jkdcbcintermediatealgebra\/wp-content\/uploads\/sites\/1021\/2020\/05\/CNX_IntAlg_Figure_10_03_312_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows the logarithmic curve going through the points (1 over 5, 1), (1, 0), and (5, negative 1).\"><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835287628\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835287630\">\n<p id=\"fs-id1167835287632\">\\(y={\\text{log}}_{0.4}x\\)<\/p>\n\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834395367\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835335823\">\n<p id=\"fs-id1167835335825\">\\(y={\\text{log}}_{0.6}x\\)<\/p>\n\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834536075\"><span data-type=\"media\" id=\"fs-id1167826825347\" data-alt=\"This figure shows the logarithmic curve going through the points (3 over 5, 1), (1, 0), and (5 over 3, negative 1).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/jkdcbcintermediatealgebra\/wp-content\/uploads\/sites\/1021\/2020\/05\/CNX_IntAlg_Figure_10_03_314_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows the logarithmic curve going through the points (3 over 5, 1), (1, 0), and (5 over 3, negative 1).\"><\/span><\/div>\n<\/div>\n<p id=\"fs-id1167831908264\"><strong data-effect=\"bold\">Solve Logarithmic Equations<\/strong><\/p>\n<p id=\"fs-id1167826998243\">In the following exercises, solve each logarithmic equation.<\/p>\n\n<div data-type=\"exercise\" id=\"fs-id1167826998246\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835375347\">\n<p id=\"fs-id1167835375350\">\\({\\text{log}}_{a}16=2\\)<\/p>\n\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835337605\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835337608\">\n<p id=\"fs-id1167835337610\">\\({\\text{log}}_{a}81=2\\)<\/p>\n\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834422749\">\n<p id=\"fs-id1167832066968\">\\(a=9\\)<\/p>\n\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835354040\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835354043\">\n<p id=\"fs-id1167835419126\">\\({\\text{log}}_{a}8=3\\)<\/p>\n\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835518006\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167828240489\">\n<p id=\"fs-id1167828240491\">\\({\\text{log}}_{a}27=3\\)<\/p>\n\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834583758\">\n<p id=\"fs-id1167834583760\">\\(a=3\\)<\/p>\n\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826864437\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834583596\">\n<p id=\"fs-id1167834583598\">\\({\\text{log}}_{a}32=2\\)<\/p>\n\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167827164772\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167830991254\">\n<p id=\"fs-id1167830991256\">\\({\\text{log}}_{a}24=3\\)<\/p>\n\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826827883\">\n<p id=\"fs-id1167826827885\">\\(a=\\sqrt[3]{24}\\)<\/p>\n\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832067469\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167832067471\">\n<p id=\"fs-id1167832067473\">\\(\\text{ln}\\phantom{\\rule{0.2em}{0ex}}x=5\\)<\/p>\n\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167827966785\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167827966787\">\n<p id=\"fs-id1167830961523\">\\(\\text{ln}\\phantom{\\rule{0.2em}{0ex}}x=4\\)<\/p>\n\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167830963537\">\n<p id=\"fs-id1167830963540\">\\(x={e}^{4}\\)<\/p>\n\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826938241\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167826938243\">\n<p id=\"fs-id1167826938246\">\\({\\text{log}}_{2}\\left(5x+1\\right)=4\\)<\/p>\n\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831887569\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167831887571\">\n<p id=\"fs-id1167831887573\">\\({\\text{log}}_{2}\\left(6x+2\\right)=5\\)<\/p>\n\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835534261\">\n<p id=\"fs-id1167835534264\">\\(x=5\\)<\/p>\n\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834300196\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834300198\">\n<p id=\"fs-id1167835283172\">\\({\\text{log}}_{3}\\left(4x-3\\right)=2\\)<\/p>\n\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834448631\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834448633\">\n<p id=\"fs-id1167834448635\">\\({\\text{log}}_{3}\\left(5x-4\\right)=4\\)<\/p>\n\n<\/div>\n<div data-type=\"solution\">\n\n\\(x=17\\)\n\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831882391\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167831882393\">\n<p id=\"fs-id1167831882395\">\\({\\text{log}}_{4}\\left(5x+6\\right)=3\\)<\/p>\n\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826967436\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167826967438\">\n\n\\({\\text{log}}_{4}\\left(3x-2\\right)=2\\)\n\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835306013\">\n<p id=\"fs-id1167835306015\">\\(x=6\\)<\/p>\n\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835303330\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835303332\">\n<p id=\"fs-id1167835303334\">\\(\\text{ln}\\phantom{\\rule{0.2em}{0ex}}{e}^{4x}=8\\)<\/p>\n\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834438873\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835381450\">\n<p id=\"fs-id1167835381452\">\\(\\text{ln}\\phantom{\\rule{0.2em}{0ex}}{e}^{2x}=6\\)<\/p>\n\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834226205\">\n<p id=\"fs-id1167834226208\">\\(x=3\\)<\/p>\n\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834429288\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834429290\">\n<p id=\"fs-id1167834429292\">\\(\\text{log}{x}^{2}=2\\)<\/p>\n\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834539458\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834539461\">\n<p id=\"fs-id1167834539463\">\\(\\text{log}\\left({x}^{2}-25\\right)=2\\)<\/p>\n\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835370937\">\n<p id=\"fs-id1167835370939\">\\(x=-5\\sqrt{5},x=5\\sqrt{5}\\)<\/p>\n\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167830837484\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167830837486\">\n<p id=\"fs-id1167830837488\">\\({\\text{log}}_{2}\\left({x}^{2}-4\\right)=5\\)<\/p>\n\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167828411060\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167828411062\">\n<p id=\"fs-id1167828411064\">\\({\\text{log}}_{3}\\left({x}^{2}+2\\right)=3\\)<\/p>\n\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835529056\">\n<p id=\"fs-id1167835529058\">\\(x=-5,x=5\\)<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id1167832042112\"><strong data-effect=\"bold\">Use Logarithmic Models in Applications<\/strong><\/p>\n<p id=\"fs-id1167832058625\">In the following exercises, use a logarithmic model to solve.<\/p>\n\n<div data-type=\"exercise\" id=\"fs-id1167832058628\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167832058631\">\n<p id=\"fs-id1167832058633\">What is the decibel level of normal conversation with intensity \\({10}^{-6}\\) watts per square inch?<\/p>\n\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826874702\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167826874704\">\n<p id=\"fs-id1167826874706\">What is the decibel level of a whisper with intensity \\({10}^{-10}\\) watts per square inch?<\/p>\n\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835390194\">\n<p id=\"fs-id1167835390196\">A whisper has a decibel level of 20 dB.<\/p>\n\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835390201\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835390204\">\n<p id=\"fs-id1167831031076\">What is the decibel level of the noise from a motorcycle with intensity \\({10}^{-2}\\) watts per square inch?<\/p>\n\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167830866003\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167830866006\">\n<p id=\"fs-id1167831811701\">What is the decibel level of the sound of a garbage disposal with intensity \\({10}^{-2}\\) watts per square inch?<\/p>\n\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835377631\">\n<p id=\"fs-id1167835377633\">The sound of a garbage disposal has a decibel level of 100 dB.<\/p>\n\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835377638\" class=\"material-set-2\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167835510386\">In 2014, Chile experienced an intense earthquake with a magnitude of \\(8.2\\) on the Richter scale. In 2010, Haiti also experienced an intense earthquake which measured \\(7.0\\) on the Richter scale. Compare the intensities of the two earthquakes.<\/p>\n\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834188881\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834188883\">\n<p id=\"fs-id1167834188886\">The Los Angeles area experiences many earthquakes. In 1994, the Northridge earthquake measured magnitude of \\(6.7\\) on the Richter scale. In 2014, Los Angeles also experienced an earthquake which measured \\(5.1\\) on the Richter scale. Compare the intensities of the two earthquakes.<\/p>\n\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835532713\">\n<p id=\"fs-id1167835532715\">The intensity of the 1994 Northridge earthquake in the Los Angeles area was about 40 times the intensity of the 2014 earthquake.<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div class=\"writing\" data-depth=\"2\" id=\"fs-id1167831112276\">\n<h4 data-type=\"title\">Writing Exercises<\/h4>\n<div data-type=\"exercise\" id=\"fs-id1167831881048\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167831881050\">\n<p id=\"fs-id1167831881052\">Explain how to change an equation from logarithmic form to exponential form.<\/p>\n\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835324862\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835324864\">\n<p id=\"fs-id1167835324867\">Explain the difference between common logarithms and natural logarithms.<\/p>\n\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835324871\">\n<p id=\"fs-id1167835524534\">Answers will vary.<\/p>\n\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835524540\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835524542\">\n<p id=\"fs-id1167835524544\">Explain why \\({\\text{log}}_{a}{a}^{x}=x.\\)<\/p>\n\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831895106\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167831895108\">\n<p id=\"fs-id1167831895110\">Explain how to find the \\({\\text{log}}_{7}32\\) on your calculator.<\/p>\n\n<\/div>\n<div data-type=\"solution\">\n<p id=\"fs-id1167835216587\">Answers will vary.<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167835216593\">\n<h4 data-type=\"title\">Self Check<\/h4>\n<p id=\"fs-id1167834188808\"><span class=\"token\">\u24d0<\/span><\/p>\n\n<div data-type=\"newline\"><\/div>\nAfter completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.<span data-type=\"media\" id=\"fs-id1167834188817\" data-alt=\"This table has four rows and five columns. The first row, which serves as a header, reads I can\u2026, Confidently, With some help, and No\u2014I don\u2019t get it. The first column below the header row reads Convert between exponential and logarithmic form, evaluate logarithmic functions, graph logarithmic functions, solve logarithmic equations, and use logarithmic models in applications. The rest of the cells are blank.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/jkdcbcintermediatealgebra\/wp-content\/uploads\/sites\/1021\/2020\/05\/CNX_IntAlg_Figure_10_03_201_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This table has four rows and five columns. The first row, which serves as a header, reads I can\u2026, Confidently, With some help, and No\u2014I don\u2019t get it. The first column below the header row reads Convert between exponential and logarithmic form, evaluate logarithmic functions, graph logarithmic functions, solve logarithmic equations, and use logarithmic models in applications. The rest of the cells are blank.\"><\/span>\n<p id=\"fs-id1167835284688\"><span class=\"token\">\u24d1<\/span> After reviewing this checklist, what will you do to become confident for all objectives?<\/p>\n\n<\/div>\n<\/div>\n<div data-type=\"glossary\" class=\"textbox shaded\">\n<h3 data-type=\"glossary-title\">Glossary<\/h3>\n<dl id=\"fs-id1167831824962\">\n \t<dt>common logarithmic function<\/dt>\n \t<dd id=\"fs-id1167831824966\">The function \\(f\\left(x\\right)=\\text{log}\\phantom{\\rule{0.2em}{0ex}}x\\) is the common logarithmic function with base\\(10,\\) where \\(x&gt;0.\\)\n<div data-type=\"newline\"><\/div>\n<div data-type=\"equation\" id=\"fs-id1167826987756\" class=\"unnumbered\" data-label=\"\">\\(y=\\text{log}\\phantom{\\rule{0.2em}{0ex}}x\\phantom{\\rule{0.2em}{0ex}}\\text{is equivalent to}\\phantom{\\rule{0.2em}{0ex}}x={10}^{y}\\)<\/div><\/dd>\n<\/dl>\n<dl id=\"fs-id1167832125662\">\n \t<dt>logarithmic function<\/dt>\n \t<dd id=\"fs-id1167834224743\">The function \\(f\\left(x\\right)={\\text{log}}_{a}x\\) is the logarithmic function with base \\(a,\\) where \\(a&gt;0,\\)\\(x&gt;0,\\) and \\(a\\ne 1.\\)\n<div data-type=\"newline\"><\/div>\n<div data-type=\"equation\" id=\"fs-id1167830770122\" class=\"unnumbered\" data-label=\"\">\\(y={\\text{log}}_{a}x\\phantom{\\rule{0.2em}{0ex}}\\text{is equivalent to}\\phantom{\\rule{0.2em}{0ex}}x={a}^{y}\\)<\/div><\/dd>\n<\/dl>\n<dl id=\"fs-id1167834464022\">\n \t<dt>natural logarithmic function<\/dt>\n \t<dd id=\"fs-id1167834464025\">The function \\(f\\left(x\\right)=\\text{ln}\\phantom{\\rule{0.2em}{0ex}}x\\) is the natural logarithmic function with base \\(e,\\) where \\(x&gt;0.\\)\n<div data-type=\"newline\"><\/div>\n<div data-type=\"equation\" id=\"fs-id1167834438819\" class=\"unnumbered\" data-label=\"\">\\(y=\\text{ln}\\phantom{\\rule{0.2em}{0ex}}x\\phantom{\\rule{0.2em}{0ex}}\\text{is equivalent to}\\phantom{\\rule{0.2em}{0ex}}x={e}^{y}\\)<\/div><\/dd>\n<\/dl>\n<\/div>","rendered":"<div class=\"textbox textbox--learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<p>By the end of this section, you will be able to:<\/p>\n<ul>\n<li>Convert between exponential and logarithmic form<\/li>\n<li>Evaluate logarithmic functions<\/li>\n<li>Graph Logarithmic functions<\/li>\n<li>Solve logarithmic equations<\/li>\n<li>Use logarithmic models in applications<\/li>\n<\/ul>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167834120941\" class=\"be-prepared\">\n<p id=\"fs-id1167835415793\">Before you get started, take this readiness quiz.<\/p>\n<ol id=\"fs-id1167834396179\" type=\"1\">\n<li>Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-1683edc2cdabb8f6fe382cfab48cf167_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#61;&#56;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"63\" style=\"vertical-align: -1px;\" \/>\n<div data-type=\"newline\"><\/div>\n<p>If you missed this problem, review <a href=\"\/contents\/da8478b4-93bc-4919-81a1-5e3267050e7e#fs-id1167836547919\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<li>Evaluate: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-a3298ae074917c4589b6f2e83a779cd7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#51;&#125;&#94;&#123;&#45;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"31\" style=\"vertical-align: 0px;\" \/>\n<div data-type=\"newline\"><\/div>\n<p>If you missed this problem, review <a href=\"\/contents\/3fa6a6c5-9a36-4dee-aea1-0166229f52fb#fs-id1167835527255\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<li>Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-04311c6ce0554cc066459b61aa9dfa0f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#50;&#125;&#94;&#123;&#52;&#125;&#61;&#51;&#120;&#45;&#53;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"94\" style=\"vertical-align: 0px;\" \/>\n<div data-type=\"newline\"><\/div>\n<p>If you missed this problem, review <a href=\"\/contents\/9f100e8f-6d15-4cae-bc22-c306e9d7d55c#fs-id1167836432956\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<\/ol>\n<\/div>\n<p>We have spent some time finding the inverse of many functions. It works well to \u2018undo\u2019 an operation with another operation. Subtracting \u2018undoes\u2019 addition, multiplication \u2018undoes\u2019 division, taking the square root \u2018undoes\u2019 squaring.<\/p>\n<p id=\"fs-id1167826880213\">As we studied the exponential function, we saw that it is one-to-one as its graphs pass the horizontal line test. This means an exponential function does have an inverse. If we try our algebraic method for finding an inverse, we run into a problem.<\/p>\n<p id=\"fs-id1167835309061\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-d9c565f296906838cc50f4d1e00907d6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#125;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#82;&#101;&#119;&#114;&#105;&#116;&#101;&#32;&#119;&#105;&#116;&#104;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#121;&#61;&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#73;&#110;&#116;&#101;&#114;&#99;&#104;&#97;&#110;&#103;&#101;&#32;&#116;&#104;&#101;&#32;&#118;&#97;&#114;&#105;&#97;&#98;&#108;&#101;&#115;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#121;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#123;&#97;&#125;&#94;&#123;&#120;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#121;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#123;&#97;&#125;&#94;&#123;&#120;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#120;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#123;&#97;&#125;&#94;&#123;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#111;&#108;&#118;&#101;&#32;&#102;&#111;&#114;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#121;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#79;&#111;&#112;&#115;&#33;&#32;&#87;&#101;&#32;&#104;&#97;&#118;&#101;&#32;&#110;&#111;&#32;&#119;&#97;&#121;&#32;&#116;&#111;&#32;&#115;&#111;&#108;&#118;&#101;&#32;&#102;&#111;&#114;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#121;&#33;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"106\" width=\"592\" style=\"vertical-align: -59px;\" \/><\/p>\n<p id=\"fs-id1167834214026\">To deal with this we define the logarithm function with base <em data-effect=\"italics\">a<\/em> to be the inverse of the exponential function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-a94bf83ed49b475db5990bf43c61d4c6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#97;&#125;&#94;&#123;&#120;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"83\" style=\"vertical-align: -4px;\" \/> We use the notation <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-06086e3c417416fef274d2f6241b01bf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#102;&#125;&#94;&#123;&#45;&#49;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#97;&#125;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"120\" style=\"vertical-align: -4px;\" \/> and say the inverse function of the exponential function is the logarithmic function.<\/p>\n<div data-type=\"note\" id=\"fs-id1167831923783\">\n<div data-type=\"title\">Logarithmic Function<\/div>\n<p id=\"fs-id1167832198559\">The function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-9b7e4e7b3caea150723d3fa578a79d72_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#97;&#125;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"102\" style=\"vertical-align: -4px;\" \/> is the <strong data-effect=\"bold\">logarithmic function<\/strong> with base <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-5c53d6ebabdbcfa4e107550ea60b1b19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\" \/>, where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-aaf802bf2187ba09e2c109dce11ba56c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#62;&#48;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"46\" style=\"vertical-align: -4px;\" \/><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-a22c7c2a68ba0f695ec4c27fcf8628d4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#62;&#48;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"47\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-31658a6c5c3f18d97495ba25506408f2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#92;&#110;&#101;&#32;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"46\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"equation\" id=\"fs-id1167834061731\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-7018593e8e8e71bf68f4cf11840968c3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#97;&#125;&#120;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#105;&#115;&#32;&#101;&#113;&#117;&#105;&#118;&#97;&#108;&#101;&#110;&#116;&#32;&#116;&#111;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#61;&#123;&#97;&#125;&#94;&#123;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"249\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167834185927\">\n<h3 data-type=\"title\">Convert Between Exponential and Logarithmic Form<\/h3>\n<p>Since the equations <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-88743ed111b206f89e0167ff135c8174_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#97;&#125;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"74\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-f8fad47c729b8835124c2ad45683c25e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#123;&#97;&#125;&#94;&#123;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"50\" style=\"vertical-align: 0px;\" \/> are equivalent, we can go back and forth between them. This will often be the method to solve some exponential and logarithmic equations. To help with converting back and forth let\u2019s take a close look at the equations. See <a href=\"#CNX_IntAlg_Figure_10_03_001\" class=\"autogenerated-content\">(Figure)<\/a>. Notice the positions of the exponent and base.<\/p>\n<div class=\"bc-figure figure\" id=\"CNX_IntAlg_Figure_10_03_001\"><span data-type=\"media\" id=\"fs-id1167834505632\" data-alt=\"This figure shows the expression y equals log sub a of x, where y is the exponent and a is the base. Next to this expression we have x equals a to the y, where again y is the exponent and a is the base.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcbcintermediatealgebra\/wp-content\/uploads\/sites\/1021\/2018\/12\/CNX_IntAlg_Figure_10_03_001.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows the expression y equals log sub a of x, where y is the exponent and a is the base. Next to this expression we have x equals a to the y, where again y is the exponent and a is the base.\" \/><\/span><\/div>\n<p id=\"fs-id1167830904022\">If we realize the logarithm is the exponent it makes the conversion easier. You may want to repeat, \u201cbase to the exponent give us the number.\u201d<\/p>\n<div data-type=\"example\" id=\"fs-id1167834528356\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167834189130\">\n<p id=\"fs-id1167835365020\">Convert to logarithmic form: <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-91ce77f25cea857e7d4ad03c9eea081f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#50;&#125;&#94;&#123;&#51;&#125;&#61;&#56;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"53\" style=\"vertical-align: -4px;\" \/> <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-7f8b73b43216dfc9dd3de7428474f30c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#53;&#125;&#94;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#125;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#53;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"71\" style=\"vertical-align: -4px;\" \/> and <span class=\"token\">\u24d2<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-a9a412d644f58a0302868b2e5cb4e910_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#120;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#49;&#54;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"79\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835479185\"><span data-type=\"media\" id=\"fs-id1167835240641\" data-alt=\"In part (a) we have 2 to the 3 power equals 8, where the 2 is red and the 3 is blue. Following this, we have blue y equals log sub red a of x. Then 3 equals log sub 2 of 8. Hence, if 2 cubed equals 8, then 3 equals log sub 2 of 8. In part (b) we have 5 to the 1 over 2 power equals square root of 5, where the 5 is red and the 1 over 2 is blue. Following this, we have blue y equals log sub red a of x. Then 1 over 2 equals log sub 5 of the square root of 5. Hence, if 5 to the 1 over 2 power equals the square root of 5, then 1 over 2 equals log sub 5 of the square root of 5. In part (c) we have 1 over 2 to the x power equals 1 over 16, where the 1 over 2 is red and the x is blue. Following this, we have blue y equals log sub red a of x. Then x equals log sub 1 over 2 of 1 over 16. Hence, if 1 over 2 to the x power equals 1 over 16, then x equals log sub 1 over 2 of 1 over 16.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcbcintermediatealgebra\/wp-content\/uploads\/sites\/1021\/2020\/05\/CNX_IntAlg_Figure_10_03_002_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"In part (a) we have 2 to the 3 power equals 8, where the 2 is red and the 3 is blue. Following this, we have blue y equals log sub red a of x. Then 3 equals log sub 2 of 8. Hence, if 2 cubed equals 8, then 3 equals log sub 2 of 8. In part (b) we have 5 to the 1 over 2 power equals square root of 5, where the 5 is red and the 1 over 2 is blue. Following this, we have blue y equals log sub red a of x. Then 1 over 2 equals log sub 5 of the square root of 5. Hence, if 5 to the 1 over 2 power equals the square root of 5, then 1 over 2 equals log sub 5 of the square root of 5. In part (c) we have 1 over 2 to the x power equals 1 over 16, where the 1 over 2 is red and the x is blue. Following this, we have blue y equals log sub red a of x. Then x equals log sub 1 over 2 of 1 over 16. Hence, if 1 over 2 to the x power equals 1 over 16, then x equals log sub 1 over 2 of 1 over 16.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167831239572\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167830703207\">\n<div data-type=\"problem\" id=\"fs-id1167835420265\">\n<p id=\"fs-id1167832056505\">Convert to logarithmic form: <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-d8c34ef5b882fda3c6d2c5efa31a54c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#51;&#125;&#94;&#123;&#50;&#125;&#61;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"49\" style=\"vertical-align: 0px;\" \/> <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-4f5b52f7f0ae12a330a80426f7b32756_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#55;&#125;&#94;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#125;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#55;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"67\" style=\"vertical-align: -2px;\" \/> <span class=\"token\">\u24d2<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-8b8a710c6e6e2aa8b445aecba19adbbf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#120;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#55;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"73\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834395261\">\n<p id=\"fs-id1167831985714\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-c67222a271e49323179b920e4965cb96_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#51;&#125;&#57;&#61;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"71\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-014dc2cd7cda751543c832ca70a6d241_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#55;&#125;&#92;&#115;&#113;&#114;&#116;&#123;&#55;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"87\" style=\"vertical-align: -6px;\" \/><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-ee2ef858298bc5f1de3317f812ef554c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#55;&#125;&#61;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"85\" style=\"vertical-align: -11px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835362910\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835347825\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167835205986\">Convert to logarithmic form: <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-1d0c9e2d24bb4bbfe20dd2637c6573fe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#52;&#125;&#94;&#123;&#51;&#125;&#61;&#54;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"58\" style=\"vertical-align: -1px;\" \/> <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-7ad27e251ab0576966378c99bd676cc4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#52;&#125;&#94;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#125;&#61;&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"68\" style=\"vertical-align: -2px;\" \/> <span class=\"token\">\u24d2<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-a5c074a1bff6460c202bfc4877afb3fc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#120;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"73\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835351385\">\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-3ed2eb9738bdb618933eeb8819681dce_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#52;&#125;&#54;&#52;&#61;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"81\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-e4034bfdf9022fc022c001e0206ead90_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#52;&#125;&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#52;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"88\" style=\"vertical-align: -6px;\" \/><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-579763d96e6cc222a1e542333f65bf79_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#50;&#125;&#61;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"85\" style=\"vertical-align: -11px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167831832871\">In the next example we do the reverse\u2014convert logarithmic form to exponential form.<\/p>\n<div data-type=\"example\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167831811617\">\n<div data-type=\"problem\" id=\"fs-id1167835233651\">\n<p id=\"fs-id1167835239496\">Convert to exponential form: <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-ccea246234d14b8ab96910ad5bde5615_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#56;&#125;&#54;&#52;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"85\" style=\"vertical-align: -4px;\" \/> <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-634a473c1d7757f8f59e3019f7bbbd4a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#52;&#125;&#49;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"76\" style=\"vertical-align: -4px;\" \/> and <span class=\"token\">\u24d2<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-c64ba413bb4793c02731b9275a9fcc58_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#49;&#48;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#49;&#48;&#48;&#48;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"118\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834064453\"><span data-type=\"media\" id=\"fs-id1167834195040\" data-alt=\"In part (a) we have 2 equals log sub 8 of 64, where the 2 is blue and the 8 is red. Following this, we have x equals red a to the blue y power. Then 64 equals 8 squared. Hence, if 2 equals log sub 8 of 64, then 64 equals 8 squared. In part (b) we have 0 equals log sub 4 of 1, where the 0 is blue and the 4 is red. Following this, we have x equals red a to the blue y power. Then 1 equals 4 to the zero power. Hence, if 0 equals log sub 4 of 1, then 1 equals 4 to the zero power. In part (c) we have negative 3 equals log sub 10 of 1 over 1000, where the negative 3 is blue and the 10 is red. Following this, we have x equals red a to the blue y power. Then 1 over 1000 equals 10 to the negative three power. Hence, if negative 3 equals log sub 10 of 1 over 1000, then 1 over 1000 equals 10 to the negative 3 power.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcbcintermediatealgebra\/wp-content\/uploads\/sites\/1021\/2020\/05\/CNX_IntAlg_Figure_10_03_003_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"In part (a) we have 2 equals log sub 8 of 64, where the 2 is blue and the 8 is red. Following this, we have x equals red a to the blue y power. Then 64 equals 8 squared. Hence, if 2 equals log sub 8 of 64, then 64 equals 8 squared. In part (b) we have 0 equals log sub 4 of 1, where the 0 is blue and the 4 is red. Following this, we have x equals red a to the blue y power. Then 1 equals 4 to the zero power. Hence, if 0 equals log sub 4 of 1, then 1 equals 4 to the zero power. In part (c) we have negative 3 equals log sub 10 of 1 over 1000, where the negative 3 is blue and the 10 is red. Following this, we have x equals red a to the blue y power. Then 1 over 1000 equals 10 to the negative three power. Hence, if negative 3 equals log sub 10 of 1 over 1000, then 1 over 1000 equals 10 to the negative 3 power.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835530849\">\n<div data-type=\"problem\" id=\"fs-id1167832195683\">\n<p>Convert to exponential form: <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-8342e9672cc61c19496d1524b537cb8a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#52;&#125;&#54;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"81\" style=\"vertical-align: -4px;\" \/> <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-8e951afcf2b19e2a4646e5d638bee4c9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#120;&#125;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"72\" style=\"vertical-align: -4px;\" \/> <span class=\"token\">\u24d2<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-a02cd2ce01154e36e0467c37f57466e7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#49;&#48;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#49;&#48;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"105\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834225090\">\n<p id=\"fs-id1167826864312\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-0273e146036efbc7c0a389e5f4d3ce0d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#52;&#61;&#123;&#52;&#125;&#94;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"57\" style=\"vertical-align: -1px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-ecc8cda197f6ecd94220c5801b7f77a2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#61;&#123;&#120;&#125;&#94;&#123;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"49\" style=\"vertical-align: -1px;\" \/><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-c67fb47af3cc3276f3dafee2d8ee8998_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#49;&#48;&#48;&#125;&#61;&#123;&#49;&#48;&#125;&#94;&#123;&#45;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"82\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167834188763\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167835302310\">Convert to exponential form: <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-436d1342e3b6d30711568de96a2c4ce5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#51;&#125;&#50;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"81\" style=\"vertical-align: -4px;\" \/> <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-8e951afcf2b19e2a4646e5d638bee4c9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#120;&#125;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"72\" style=\"vertical-align: -4px;\" \/> <span class=\"token\">\u24d2<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-35a9abea1932f1a84b28b62dc1572de0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#49;&#48;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#49;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"98\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826986943\">\n<p id=\"fs-id1167834462881\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-12f80419f034a46a385136274e0a350c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#55;&#61;&#123;&#51;&#125;&#94;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"57\" style=\"vertical-align: 0px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-ecc8cda197f6ecd94220c5801b7f77a2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#61;&#123;&#120;&#125;&#94;&#123;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"49\" style=\"vertical-align: -1px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-5b68ae5b45e6d754855604889184dd50_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#49;&#48;&#125;&#61;&#123;&#49;&#48;&#125;&#94;&#123;&#45;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"74\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167834537381\">\n<h3 data-type=\"title\">Evaluate Logarithmic Functions<\/h3>\n<p id=\"fs-id1167830837011\">We can solve and evaluate logarithmic equations by using the technique of converting the equation to its equivalent exponential equation.<\/p>\n<div data-type=\"example\" id=\"fs-id1167826978563\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167835218009\">\n<div data-type=\"problem\" id=\"fs-id1167834505593\">\n<p id=\"fs-id1167835306719\">Find the value of <em data-effect=\"italics\">x<\/em>: <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-567887c0ea09b954fc901a0b6c91717c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#120;&#125;&#51;&#54;&#61;&#50;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"86\" style=\"vertical-align: -4px;\" \/> <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-ed44e2725b9573be9e38d396b6de0115_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#52;&#125;&#120;&#61;&#51;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"77\" style=\"vertical-align: -4px;\" \/> and <span class=\"token\">\u24d2<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-4445691c9a8a694d7c0b1977db876762_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#56;&#125;&#61;&#120;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"82\" style=\"vertical-align: -11px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834431960\">\n<p id=\"fs-id1167834535355\"><span class=\"token\">\u24d0<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-9e8ea3f9bf724ce7194d2f10b2428f39_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#46;&#56;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#120;&#125;&#51;&#54;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#61;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#111;&#110;&#118;&#101;&#114;&#116;&#32;&#116;&#111;&#32;&#101;&#120;&#112;&#111;&#110;&#101;&#110;&#116;&#105;&#97;&#108;&#32;&#102;&#111;&#114;&#109;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#53;&#46;&#54;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#61;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#51;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#111;&#108;&#118;&#101;&#32;&#116;&#104;&#101;&#32;&#113;&#117;&#97;&#100;&#114;&#97;&#116;&#105;&#99;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#46;&#56;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#61;&#54;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#111;&#118;&#101;&#114;&#108;&#105;&#110;&#101;&#123;&#41;&#120;&#61;&#45;&#54;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#104;&#101;&#32;&#98;&#97;&#115;&#101;&#32;&#111;&#102;&#32;&#97;&#32;&#108;&#111;&#103;&#97;&#114;&#105;&#116;&#104;&#109;&#105;&#99;&#32;&#102;&#117;&#110;&#99;&#116;&#105;&#111;&#110;&#32;&#109;&#117;&#115;&#116;&#32;&#98;&#101;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#112;&#111;&#115;&#105;&#116;&#105;&#118;&#101;&#44;&#32;&#115;&#111;&#32;&#119;&#101;&#32;&#101;&#108;&#105;&#109;&#105;&#110;&#97;&#116;&#101;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#61;&#45;&#54;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#54;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#61;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#54;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#104;&#101;&#114;&#101;&#102;&#111;&#114;&#101;&#44;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#54;&#125;&#51;&#54;&#61;&#50;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"107\" width=\"756\" style=\"vertical-align: -49px;\" \/><\/p>\n<p id=\"fs-id1167831040444\"><span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-65284448539453808339ebebf1d4f0c1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#57;&#46;&#54;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#52;&#125;&#120;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#111;&#110;&#118;&#101;&#114;&#116;&#32;&#116;&#111;&#32;&#101;&#120;&#112;&#111;&#110;&#101;&#110;&#116;&#105;&#97;&#108;&#32;&#102;&#111;&#114;&#109;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#56;&#46;&#54;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#52;&#125;&#94;&#123;&#51;&#125;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#120;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#105;&#109;&#112;&#108;&#105;&#102;&#121;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#56;&#46;&#54;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#54;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#104;&#101;&#114;&#101;&#102;&#111;&#114;&#101;&#44;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#52;&#125;&#54;&#52;&#61;&#51;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"83\" width=\"549\" style=\"vertical-align: -37px;\" \/><\/p>\n<p id=\"fs-id1167828395816\"><span class=\"token\">\u24d2<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<pre class=\"ql-errors\">*** QuickLaTeX cannot compile formula:\n&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#125;&#92;&#92;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#56;&#125;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#120;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#111;&#110;&#118;&#101;&#114;&#116;&#32;&#116;&#111;&#32;&#101;&#120;&#112;&#111;&#110;&#101;&#110;&#116;&#105;&#97;&#108;&#32;&#102;&#111;&#114;&#109;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#120;&#125;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#56;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#82;&#101;&#119;&#114;&#105;&#116;&#101;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#56;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#115;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#51;&#125;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#120;&#125;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#51;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#87;&#105;&#116;&#104;&#32;&#116;&#104;&#101;&#32;&#115;&#97;&#109;&#101;&#32;&#98;&#97;&#115;&#101;&#44;&#32;&#116;&#104;&#101;&#32;&#101;&#120;&#112;&#111;&#110;&#101;&#110;&#116;&#115;&#32;&#109;&#117;&#115;&#116;&#32;&#98;&#101;&#32;&#101;&#113;&#117;&#97;&#108;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#120;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#104;&#101;&#114;&#101;&#102;&#111;&#114;&#101;&#44;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#56;&#125;&#61;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\n\n*** Error message:\n&#77;&#105;&#115;&#115;&#105;&#110;&#103;&#32;&#35;&#32;&#105;&#110;&#115;&#101;&#114;&#116;&#101;&#100;&#32;&#105;&#110;&#32;&#97;&#108;&#105;&#103;&#110;&#109;&#101;&#110;&#116;&#32;&#112;&#114;&#101;&#97;&#109;&#98;&#108;&#101;&#46;\r\n&#108;&#101;&#97;&#100;&#105;&#110;&#103;&#32;&#116;&#101;&#120;&#116;&#58;&#32;&#36;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#125;\r\n&#69;&#120;&#116;&#114;&#97;&#32;&#97;&#108;&#105;&#103;&#110;&#109;&#101;&#110;&#116;&#32;&#116;&#97;&#98;&#32;&#104;&#97;&#115;&#32;&#98;&#101;&#101;&#110;&#32;&#99;&#104;&#97;&#110;&#103;&#101;&#100;&#32;&#116;&#111;&#32;&#92;&#99;&#114;&#46;\r\n&#108;&#101;&#97;&#100;&#105;&#110;&#103;&#32;&#116;&#101;&#120;&#116;&#58;&#32;&#36;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#125;&#92;&#92;&#32;&#92;&#92;&#32;&#38;\r\n&#69;&#120;&#116;&#114;&#97;&#32;&#97;&#108;&#105;&#103;&#110;&#109;&#101;&#110;&#116;&#32;&#116;&#97;&#98;&#32;&#104;&#97;&#115;&#32;&#98;&#101;&#101;&#110;&#32;&#99;&#104;&#97;&#110;&#103;&#101;&#100;&#32;&#116;&#111;&#32;&#92;&#99;&#114;&#46;\r\n&#108;&#101;&#97;&#100;&#105;&#110;&#103;&#32;&#116;&#101;&#120;&#116;&#58;&#32;&#36;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#125;&#92;&#92;&#32;&#92;&#92;&#32;&#38;&#32;&#38;\r\n&#69;&#120;&#116;&#114;&#97;&#32;&#97;&#108;&#105;&#103;&#110;&#109;&#101;&#110;&#116;&#32;&#116;&#97;&#98;&#32;&#104;&#97;&#115;&#32;&#98;&#101;&#101;&#110;&#32;&#99;&#104;&#97;&#110;&#103;&#101;&#100;&#32;&#116;&#111;&#32;&#92;&#99;&#114;&#46;\r\n&#108;&#101;&#97;&#100;&#105;&#110;&#103;&#32;&#116;&#101;&#120;&#116;&#58;&#32;&#36;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#125;&#92;&#92;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;\r\n&#77;&#105;&#115;&#115;&#105;&#110;&#103;&#32;&#36;&#32;&#105;&#110;&#115;&#101;&#114;&#116;&#101;&#100;&#46;\r\n&#108;&#101;&#97;&#100;&#105;&#110;&#103;&#32;&#116;&#101;&#120;&#116;&#58;&#32;&#46;&#46;&#46;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;\r\n&#77;&#105;&#115;&#115;&#105;&#110;&#103;&#32;&#36;&#32;&#105;&#110;&#115;&#101;&#114;&#116;&#101;&#100;&#46;\r\n&#108;&#101;&#97;&#100;&#105;&#110;&#103;&#32;&#116;&#101;&#120;&#116;&#58;&#32;&#46;&#46;&#46;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#56;&#125;&#38;\r\n&#69;&#120;&#116;&#114;&#97;&#32;&#97;&#108;&#105;&#103;&#110;&#109;&#101;&#110;&#116;&#32;&#116;&#97;&#98;&#32;&#104;&#97;&#115;&#32;&#98;&#101;&#101;&#110;&#32;&#99;&#104;&#97;&#110;&#103;&#101;&#100;&#32;&#116;&#111;&#32;&#92;&#99;&#114;&#46;\r\n&#108;&#101;&#97;&#100;&#105;&#110;&#103;&#32;&#116;&#101;&#120;&#116;&#58;&#32;&#46;&#46;&#46;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#56;&#125;&#38;\r\n&#69;&#120;&#116;&#114;&#97;&#32;&#97;&#108;&#105;&#103;&#110;&#109;&#101;&#110;&#116;&#32;&#116;&#97;&#98;&#32;&#104;&#97;&#115;&#32;&#98;&#101;&#101;&#110;&#32;&#99;&#104;&#97;&#110;&#103;&#101;&#100;&#32;&#116;&#111;&#32;&#92;&#99;&#114;&#46;\r\n&#108;&#101;&#97;&#100;&#105;&#110;&#103;&#32;&#116;&#101;&#120;&#116;&#58;&#32;&#46;&#46;&#46;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#56;&#125;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;\r\n&#69;&#120;&#116;&#114;&#97;&#32;&#97;&#108;&#105;&#103;&#110;&#109;&#101;&#110;&#116;&#32;&#116;&#97;&#98;&#32;&#104;&#97;&#115;&#32;&#98;&#101;&#101;&#110;&#32;&#99;&#104;&#97;&#110;&#103;&#101;&#100;&#32;&#116;&#111;&#32;&#92;&#99;&#114;&#46;\r\n&#108;&#101;&#97;&#100;&#105;&#110;&#103;&#32;&#116;&#101;&#120;&#116;&#58;&#32;&#46;&#46;&#46;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#111;&#110;&#118;&#101;&#114;&#116;&#32;&#116;&#111;&#32;&#101;&#120;&#112;&#111;&#110;&#101;&#110;&#116;&#105;&#97;&#108;&#32;&#102;&#111;&#114;&#109;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;\r\n\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167832044030\">\n<div data-type=\"problem\" id=\"fs-id1167835216027\">\n<p id=\"fs-id1167835324805\">Find the value of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-845f2902b8bebf60c3c7372a7fbe4d02_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#58;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"19\" style=\"vertical-align: 0px;\" \/> <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-f829e267f7340a78d0c38cbc0caaf101_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#120;&#125;&#54;&#52;&#61;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"81\" style=\"vertical-align: -4px;\" \/> <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-8617f5a993d41603738d8873d1e55208_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#53;&#125;&#120;&#61;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"73\" style=\"vertical-align: -4px;\" \/> <span class=\"token\">\u24d2<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-1e9222dbeaf92e43723c389420d1423d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#61;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"78\" style=\"vertical-align: -11px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-1dc438da70f358c2eb1bf64a8b7ea4b5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"43\" style=\"vertical-align: 0px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-bd200032faab729d3376085fad9115bc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#49;&#50;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"60\" style=\"vertical-align: -1px;\" \/><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-c657687cbbf5ea9a7545edb42190e592_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"42\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167834099089\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835337372\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167835307439\">Find the value of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-845f2902b8bebf60c3c7372a7fbe4d02_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#58;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"19\" style=\"vertical-align: 0px;\" \/> <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-d1f8860760191f7fd8bfd25de08fe664_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#120;&#125;&#56;&#49;&#61;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"81\" style=\"vertical-align: -4px;\" \/> <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-957598484518c44dfbc4b2a62c3f1694_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#51;&#125;&#120;&#61;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"72\" style=\"vertical-align: -4px;\" \/> <span class=\"token\">\u24d2<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-ee2ef858298bc5f1de3317f812ef554c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#55;&#125;&#61;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"85\" style=\"vertical-align: -11px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835351129\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-300a345ef7b973d34879ac8e90555390_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"43\" style=\"vertical-align: 0px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-bc50cb85a78e60556cec12af7a4da05a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#50;&#52;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"61\" style=\"vertical-align: -1px;\" \/><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-3573bf1ea4c223bb71878796b2106731_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"43\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167830699532\">When see an expression such as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-8cb5296647d4f6fc9618b92d6ca5c7d7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#51;&#125;&#50;&#55;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"52\" style=\"vertical-align: -4px;\" \/> we can find its exact value two ways. By inspection we realize it means <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-0b51fef7d4f134f7ca9c69005e7cd94c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#96;&#96;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"16\" style=\"vertical-align: 0px;\" \/> to what power will be <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-620f16ecb5c1fac11e64495f8c6d103c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#55;&#39;&#39;&#63;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"34\" style=\"vertical-align: 0px;\" \/> Since <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-77a47e6db7891837b1aafe75f65e8f97_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#51;&#125;&#94;&#123;&#51;&#125;&#61;&#50;&#55;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"62\" style=\"vertical-align: -4px;\" \/> we know <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-00a7d147fb3d0cfffdb5f24be177eea6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#51;&#125;&#50;&#55;&#61;&#51;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"85\" style=\"vertical-align: -4px;\" \/> An alternate way is to set the expression equal to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/> and then convert it into an exponential equation.<\/p>\n<div data-type=\"example\" id=\"fs-id1167834501185\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167835534392\">\n<p id=\"fs-id1167835512014\">Find the exact value of each logarithm without using a calculator:<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-0738fd5d162b820dd2ed141007abb8dd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#53;&#125;&#50;&#53;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"52\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-f864b963f4f7bc7cf2e50a625fbcf32d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#57;&#125;&#51;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"43\" style=\"vertical-align: -4px;\" \/> and <span class=\"token\">\u24d2<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-228e977fdca4a772cb891000670388f1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#50;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#49;&#54;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"52\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167832126125\">\n<p id=\"fs-id1167832052302\"><span class=\"token\">\u24d0<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-91d5b6148fdb8986b20256d166de1a83_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#53;&#125;&#50;&#53;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#53;&#32;&#116;&#111;&#32;&#119;&#104;&#97;&#116;&#32;&#112;&#111;&#119;&#101;&#114;&#32;&#119;&#105;&#108;&#108;&#32;&#98;&#101;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#50;&#53;&#63;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#53;&#125;&#50;&#53;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#79;&#114;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#101;&#116;&#32;&#116;&#104;&#101;&#32;&#101;&#120;&#112;&#114;&#101;&#115;&#115;&#105;&#111;&#110;&#32;&#101;&#113;&#117;&#97;&#108;&#32;&#116;&#111;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#53;&#125;&#50;&#53;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#120;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#104;&#97;&#110;&#103;&#101;&#32;&#116;&#111;&#32;&#101;&#120;&#112;&#111;&#110;&#101;&#110;&#116;&#105;&#97;&#108;&#32;&#102;&#111;&#114;&#109;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#53;&#125;&#94;&#123;&#120;&#125;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#50;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#82;&#101;&#119;&#114;&#105;&#116;&#101;&#32;&#50;&#53;&#32;&#97;&#115;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#53;&#125;&#94;&#123;&#50;&#125;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#53;&#125;&#94;&#123;&#120;&#125;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#123;&#53;&#125;&#94;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#87;&#105;&#116;&#104;&#32;&#116;&#104;&#101;&#32;&#115;&#97;&#109;&#101;&#32;&#98;&#97;&#115;&#101;&#32;&#116;&#104;&#101;&#32;&#101;&#120;&#112;&#111;&#110;&#101;&#110;&#116;&#115;&#32;&#109;&#117;&#115;&#116;&#32;&#98;&#101;&#32;&#101;&#113;&#117;&#97;&#108;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#120;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#104;&#101;&#114;&#101;&#102;&#111;&#114;&#101;&#44;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#53;&#125;&#50;&#53;&#61;&#50;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"171\" width=\"548\" style=\"vertical-align: -81px;\" \/><\/p>\n<p><span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-49eefd11abeaef7e06feb2cc43581bfe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#57;&#125;&#51;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#101;&#116;&#32;&#116;&#104;&#101;&#32;&#101;&#120;&#112;&#114;&#101;&#115;&#115;&#105;&#111;&#110;&#32;&#101;&#113;&#117;&#97;&#108;&#32;&#116;&#111;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#57;&#125;&#51;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#120;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#104;&#97;&#110;&#103;&#101;&#32;&#116;&#111;&#32;&#101;&#120;&#112;&#111;&#110;&#101;&#110;&#116;&#105;&#97;&#108;&#32;&#102;&#111;&#114;&#109;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#57;&#125;&#94;&#123;&#120;&#125;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#82;&#101;&#119;&#114;&#105;&#116;&#101;&#32;&#57;&#32;&#97;&#115;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#51;&#125;&#94;&#123;&#50;&#125;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#51;&#125;&#94;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#120;&#125;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#123;&#51;&#125;&#94;&#123;&#49;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#105;&#109;&#112;&#108;&#105;&#102;&#121;&#32;&#116;&#104;&#101;&#32;&#101;&#120;&#112;&#111;&#110;&#101;&#110;&#116;&#115;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#51;&#125;&#94;&#123;&#50;&#120;&#125;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#123;&#51;&#125;&#94;&#123;&#49;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#87;&#105;&#116;&#104;&#32;&#116;&#104;&#101;&#32;&#115;&#97;&#109;&#101;&#32;&#98;&#97;&#115;&#101;&#32;&#116;&#104;&#101;&#32;&#101;&#120;&#112;&#111;&#110;&#101;&#110;&#116;&#115;&#32;&#109;&#117;&#115;&#116;&#32;&#98;&#101;&#32;&#101;&#113;&#117;&#97;&#108;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#50;&#120;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#111;&#108;&#118;&#101;&#32;&#116;&#104;&#101;&#32;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#120;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#104;&#101;&#114;&#101;&#102;&#111;&#114;&#101;&#44;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#57;&#125;&#51;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"173\" width=\"539\" style=\"vertical-align: -83px;\" \/><\/p>\n<p id=\"fs-id1167834593440\"><span class=\"token\">\u24d2<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-51a48e3e0cb51d54a01c8143b88fd9b5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#50;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#49;&#54;&#125;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#101;&#116;&#32;&#116;&#104;&#101;&#32;&#101;&#120;&#112;&#114;&#101;&#115;&#115;&#105;&#111;&#110;&#32;&#101;&#113;&#117;&#97;&#108;&#32;&#116;&#111;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#50;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#49;&#54;&#125;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#120;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#104;&#97;&#110;&#103;&#101;&#32;&#116;&#111;&#32;&#101;&#120;&#112;&#111;&#110;&#101;&#110;&#116;&#105;&#97;&#108;&#32;&#102;&#111;&#114;&#109;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#50;&#125;&#94;&#123;&#120;&#125;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#49;&#54;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#82;&#101;&#119;&#114;&#105;&#116;&#101;&#32;&#49;&#54;&#32;&#97;&#115;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#50;&#125;&#94;&#123;&#52;&#125;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#50;&#125;&#94;&#123;&#120;&#125;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#123;&#50;&#125;&#94;&#123;&#52;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#50;&#125;&#94;&#123;&#120;&#125;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#123;&#50;&#125;&#94;&#123;&#45;&#52;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#87;&#105;&#116;&#104;&#32;&#116;&#104;&#101;&#32;&#115;&#97;&#109;&#101;&#32;&#98;&#97;&#115;&#101;&#32;&#116;&#104;&#101;&#32;&#101;&#120;&#112;&#111;&#110;&#101;&#110;&#116;&#115;&#32;&#109;&#117;&#115;&#116;&#32;&#98;&#101;&#32;&#101;&#113;&#117;&#97;&#108;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#120;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#45;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#104;&#101;&#114;&#101;&#102;&#111;&#114;&#101;&#44;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#50;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#49;&#54;&#125;&#61;&#45;&#52;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"155\" width=\"558\" style=\"vertical-align: -73px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" class=\"try\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167834377442\">\n<p id=\"fs-id1167831823950\">Find the exact value of each logarithm without using a calculator:<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-700c4e7c74c35735158b937145b2c28f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#49;&#50;&#125;&#49;&#52;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"64\" style=\"vertical-align: -5px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-35f212fa3aba6e811dc756b19280c6e0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#52;&#125;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-d63affefa67f1376e339d783cff7e45a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#50;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"46\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834556149\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p>2 <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-b8fa03e1b526c6d07ec843385490ca4c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/> <span class=\"token\">\u24d2<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-7b5b9d9f382b11767d19f257afca0019_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"21\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167834346951\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167834464088\">\n<div data-type=\"problem\" id=\"fs-id1167834299816\">\n<p id=\"fs-id1167834547189\">Find the exact value of each logarithm without using a calculator:<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-81182120a09ad317f3aee2c2c0fe3b92_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#57;&#125;&#56;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"47\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-d3fa18898107d9ae9747cd864ef2f259_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#56;&#125;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-5d9067e90a97facd9d4fda21163ef8f4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#51;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"39\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834537147\">\n<p><span class=\"token\">\u24d0<\/span> 2 <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-34f0e870957984f6c69249b8cf4f5813_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/> <span class=\"token\">\u24d2<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-17c33e2329e29a62a80ad2b547b4753d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"21\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167835233646\">\n<h3 data-type=\"title\">Graph Logarithmic Functions<\/h3>\n<p id=\"fs-id1167835306032\">To graph a logarithmic function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-da9c921b31ae1aeb9123bc7f2f6a0c3a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#97;&#125;&#120;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"78\" style=\"vertical-align: -4px;\" \/> it is easiest to convert the equation to its exponential form, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-8cd922db315a7363a57654e04cff3ba8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#123;&#97;&#125;&#94;&#123;&#121;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"55\" style=\"vertical-align: 0px;\" \/> Generally, when we look for ordered pairs for the graph of a function, we usually choose an <em data-effect=\"italics\">x<\/em>-value and then determine its corresponding <em data-effect=\"italics\">y<\/em>-value. In this case you may find it easier to choose <em data-effect=\"italics\">y<\/em>-values and then determine its corresponding <em data-effect=\"italics\">x<\/em>-value.<\/p>\n<div data-type=\"example\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167832059991\">\n<div data-type=\"problem\" id=\"fs-id1167831071344\">\n<p id=\"fs-id1167832066288\">Graph <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-b2839ec31180416dfb3493c5afd54cf3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#50;&#125;&#120;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"77\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826849526\">\n<p id=\"fs-id1167831239155\">To graph the function, we will first rewrite the logarithmic equation, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-e5740010383f6bc944497db93968ffb8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#50;&#125;&#120;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"77\" style=\"vertical-align: -4px;\" \/> in exponential form, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-a443db2ba4f5b839b4ca156ccdd9ba8f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#50;&#125;&#94;&#123;&#121;&#125;&#61;&#120;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"55\" style=\"vertical-align: 0px;\" \/><\/p>\n<p id=\"fs-id1167826994962\">We will use point plotting to graph the function. It will be easier to start with values of <em data-effect=\"italics\">y<\/em> and then get <em data-effect=\"italics\">x<\/em>.<\/p>\n<table class=\"unnumbered\" summary=\"This table has three columns and seven rows. The first row is a header row and it reads y, 2 to the y power equals x, and (x, y). In the first column below y we have negative 2, negative 1, 0, 1, 2, and 3. In the second column below 2 to the y power equals x we have 2 to the negative 2 power equals 1 over 2 squared which equals 1 over 4, 2 to the negative 1 power equals 1 over 2 to the first power which equals 1 over 2, 2 to the negative 0 power equals 2, 2 to the 1 power equals 2, 2 squared equals 4, and 2 cubed equals 8. In the third column below (x, y) we have (1 over 4, 2), (1 over 2, negative 1), (1, 0), (2, 1), (4, 2), and (8, 3).\" data-label=\"\">\n<thead>\n<tr>\n<th data-valign=\"middle\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-0af556714940c351c933bba8cf840796_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\" \/><\/th>\n<th data-valign=\"middle\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-48b6401f132cf9082e365efdc40033cb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#50;&#125;&#94;&#123;&#121;&#125;&#61;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"51\" style=\"vertical-align: 0px;\" \/><\/th>\n<th data-valign=\"middle\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-aee61752ae042431152087f74b766103_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#44;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"39\" style=\"vertical-align: -4px;\" \/><\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td data-valign=\"middle\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-17c33e2329e29a62a80ad2b547b4753d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"21\" style=\"vertical-align: 0px;\" \/><\/td>\n<td data-valign=\"middle\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-7dfa4b26520df97bf43c738562a17930_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#50;&#125;&#94;&#123;&#45;&#50;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#123;&#50;&#125;&#94;&#123;&#50;&#125;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"101\" style=\"vertical-align: -7px;\" \/><\/td>\n<td data-valign=\"middle\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-09b0d6886aaecd20786f7acc190e14e3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#44;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"40\" style=\"vertical-align: -7px;\" \/><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"middle\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-7b34c01098c83fa602de54e9d74d63a9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"21\" style=\"vertical-align: -1px;\" \/><\/td>\n<td data-valign=\"middle\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-a852b54cf3a620448dd0c7fa52dd3739_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#50;&#125;&#94;&#123;&#45;&#49;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#123;&#50;&#125;&#94;&#123;&#49;&#125;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"101\" style=\"vertical-align: -7px;\" \/><\/td>\n<td data-valign=\"middle\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-95b5a5084adb2eb5c8c447c41c82b93e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#44;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"54\" style=\"vertical-align: -7px;\" \/><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"middle\" data-align=\"left\">0<\/td>\n<td data-valign=\"middle\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-3e5553b60802088fd831b029bc495768_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#50;&#125;&#94;&#123;&#48;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"48\" style=\"vertical-align: -1px;\" \/><\/td>\n<td data-valign=\"middle\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-d6c757b148db3a6707e7e1f199d80f9f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"middle\" data-align=\"left\">1<\/td>\n<td data-valign=\"middle\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-e7f47dfb47302a369c969f721179e542_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#50;&#125;&#94;&#123;&#49;&#125;&#61;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"48\" style=\"vertical-align: 0px;\" \/><\/td>\n<td data-valign=\"middle\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-bb160c5e6177bdd7a1d220c410258e5f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"middle\" data-align=\"left\">2<\/td>\n<td data-valign=\"middle\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-fa9f33eed7fe67e4c9b6e4c6e6d4ddbb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#50;&#125;&#94;&#123;&#50;&#125;&#61;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"49\" style=\"vertical-align: -1px;\" \/><\/td>\n<td data-valign=\"middle\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-eb6560ec04491f4e7cf02e14e6df5ec3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#44;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">3<\/td>\n<td data-valign=\"middle\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-6a2dc3ae03aca55b85f1000122162e0d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#50;&#125;&#94;&#123;&#51;&#125;&#61;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"49\" style=\"vertical-align: 0px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-38f47676ff4305b6e1df48d364862ed0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#56;&#44;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><span data-type=\"media\" id=\"fs-id1167835328463\" data-alt=\"This figure shows the logarithmic curve going through the points (1 over 2, negative 1), (1, 0), and (2, 1).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcbcintermediatealgebra\/wp-content\/uploads\/sites\/1021\/2020\/05\/CNX_IntAlg_Figure_10_03_004_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows the logarithmic curve going through the points (1 over 2, negative 1), (1, 0), and (2, 1).\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835201098\" class=\"try\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167831116958\">Graph: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-dc99c777a7d309a4e0212a883757c69a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#51;&#125;&#120;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"77\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" data-alt=\"This figure shows the logarithmic curve going through the points (1 over 3, negative 1), (1, 0), and (3, 1).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcbcintermediatealgebra\/wp-content\/uploads\/sites\/1021\/2020\/05\/CNX_IntAlg_Figure_10_03_301_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows the logarithmic curve going through the points (1 over 3, negative 1), (1, 0), and (3, 1).\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" class=\"try\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167831117010\">Graph: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-2e8de700aba5dc32771a0911cc5ab691_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#53;&#125;&#120;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"77\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167835336536\" data-alt=\"This figure shows the logarithmic curve going through the points (1 over 5, negative 1), (1, 0), and (5, 1).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcbcintermediatealgebra\/wp-content\/uploads\/sites\/1021\/2020\/05\/CNX_IntAlg_Figure_10_03_302_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows the logarithmic curve going through the points (1 over 5, negative 1), (1, 0), and (5, 1).\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167831191366\">The graphs of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-e5740010383f6bc944497db93968ffb8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#50;&#125;&#120;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"77\" style=\"vertical-align: -4px;\" \/><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-23dd1bf4a3d0127d09af9f08754c5596_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#51;&#125;&#120;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"77\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-dcf95a39f891bcdf8ad06ef5ff212a76_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#53;&#125;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"73\" style=\"vertical-align: -4px;\" \/> are the shape we expect from a logarithmic function where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-e81ef2f32f10882cbba77d80a157a26e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#62;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"46\" style=\"vertical-align: -1px;\" \/><\/p>\n<p id=\"fs-id1167835361417\">We notice that for each function the graph contains the point <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-579e1f7c2d61eab1dffcc2d5d02de664_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"45\" style=\"vertical-align: -4px;\" \/> This make sense because <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-656872228fc6ec0391842cc6e945a795_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#97;&#125;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"71\" style=\"vertical-align: -4px;\" \/> means <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-1001cb2e5ed58ec551bbd3d4da4e0b0e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#94;&#123;&#48;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"49\" style=\"vertical-align: -1px;\" \/> which is true for any <em data-effect=\"italics\">a<\/em>.<\/p>\n<p id=\"fs-id1167834527903\">The graph of each function, also contains the point <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-8dbe1b21f6264d640e0f268a40caa52c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#44;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"46\" style=\"vertical-align: -4px;\" \/> This makes sense as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-8250c991ad293e5949392bf7e3f4af7b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#97;&#125;&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"71\" style=\"vertical-align: -4px;\" \/> means <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-9754f701dfb87248bdc3742c17a3edb1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#94;&#123;&#49;&#125;&#61;&#97;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"54\" style=\"vertical-align: 0px;\" \/> which is true for any <em data-effect=\"italics\">a<\/em>.<\/p>\n<p id=\"fs-id1167835173724\">Notice too, the graph of each function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-88743ed111b206f89e0167ff135c8174_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#97;&#125;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"74\" style=\"vertical-align: -4px;\" \/> also contains the point <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-6a1f5e5ad1e963280b4e34145679a907_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#97;&#125;&#44;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"63\" style=\"vertical-align: -7px;\" \/> This makes sense as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-ad4d521904824591062e1c40a546542a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#97;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#97;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"86\" style=\"vertical-align: -6px;\" \/> means <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-976773417e3462c1c6d75381245d9a7f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#94;&#123;&#45;&#49;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#97;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"67\" style=\"vertical-align: -6px;\" \/> which is true for any <em data-effect=\"italics\">a<\/em>.<\/p>\n<p id=\"fs-id1167835308174\">Look at each graph again. Now we will see that many characteristics of the logarithm function are simply \u2019mirror images\u2019 of the characteristics of the corresponding exponential function.<\/p>\n<p>What is the domain of the function? The graph never hits the <em data-effect=\"italics\">y<\/em>-axis. The domain is all positive numbers. We write the domain in interval notation as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-8a9111b3c516b8210661ecc5f4be0f2d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#92;&#105;&#110;&#102;&#116;&#121;&#32;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"54\" style=\"vertical-align: -4px;\" \/><\/p>\n<p id=\"fs-id1167826804614\">What is the range for each function? From the graphs we can see that the range is the set of all real numbers. There is no restriction on the range. We write the range in interval notation as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-5df92a2cc0a08c383e86c9bcbca032bf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#105;&#110;&#102;&#116;&#121;&#32;&#44;&#92;&#105;&#110;&#102;&#116;&#121;&#32;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"63\" style=\"vertical-align: -4px;\" \/><\/p>\n<p id=\"fs-id1167835423129\">When the graph approaches the <em data-effect=\"italics\">y<\/em>-axis so very closely but will never cross it, we call the line <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-534b43efb5ec72c9aa9f5eaccec09e9c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#48;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"47\" style=\"vertical-align: -4px;\" \/> the <em data-effect=\"italics\">y<\/em>-axis, a vertical asymptote.<\/p>\n<div data-type=\"note\" id=\"fs-id1167830705415\">\n<div data-type=\"title\">Properties of the Graph of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-88743ed111b206f89e0167ff135c8174_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#97;&#125;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"74\" style=\"vertical-align: -4px;\" \/> when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-eb7aa260ff031e44ade9124e9760983a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#62;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"41\" style=\"vertical-align: -1px;\" \/><\/div>\n<table class=\"unnumbered\" summary=\"Table has two columns and six rows. The first row shows the domain is 0 to infinity. The second row shows the range is negative infinity to infinity. The third row shows the x intercept is 1, 0. The fourth row shows there is no y-intercept. The fifth row shows the function contains a, 1 and 1 over a, negative 1. The sixth column shows the asymptote is the y axis.\" data-label=\"\">\n<tbody>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Domain<\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-264878915584580308929a53513b9f1f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#92;&#105;&#110;&#102;&#116;&#121;&#32;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"47\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Range<\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-90eaa2f0f55290692476556ec134a082_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#105;&#110;&#102;&#116;&#121;&#32;&#44;&#92;&#105;&#110;&#102;&#116;&#121;&#32;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"56\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-b54c0b63a237faaaeef5ef6ae9778b5b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#105;&#110;&#116;&#101;&#114;&#99;&#101;&#112;&#116;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"84\" style=\"vertical-align: -3px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-d6c757b148db3a6707e7e1f199d80f9f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-2f818976d51dc656017a8ff9cf146d07_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#105;&#110;&#116;&#101;&#114;&#99;&#101;&#112;&#116;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"84\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\">None<\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Contains<\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-ca70a98fd69d9bc6c79a48c493db95b5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#44;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"46\" style=\"vertical-align: -4px;\" \/><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-5ef51d1178752973f3de098e0149e761_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#97;&#125;&#44;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"54\" style=\"vertical-align: -7px;\" \/><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Asymptote<\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-5fd77bea1da9dad460894d29cd0209d0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#120;&#105;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"45\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><span data-type=\"media\" data-alt=\"This figure shows the logarithmic curve going through the points (1 over a, negative 1), (1, 0), and (a, 1).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcbcintermediatealgebra\/wp-content\/uploads\/sites\/1021\/2020\/05\/CNX_IntAlg_Figure_10_03_005_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows the logarithmic curve going through the points (1 over a, negative 1), (1, 0), and (a, 1).\" \/><\/span><\/p>\n<\/div>\n<p>Our next example looks at the graph of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-88743ed111b206f89e0167ff135c8174_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#97;&#125;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"74\" style=\"vertical-align: -4px;\" \/> when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-ee823d38f20b11152c6360028dfe63e4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#60;&#97;&#60;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"79\" style=\"vertical-align: -1px;\" \/><\/p>\n<div data-type=\"example\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167835517856\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167835230729\">Graph <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-ed632b9c752be4dfa9c44455fe6457e3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#125;&#120;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"80\" style=\"vertical-align: -11px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834059251\">\n<p>To graph the function, we will first rewrite the logarithmic equation, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-3d597d65569a1779e9380ca6ec65edd7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#125;&#120;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"80\" style=\"vertical-align: -11px;\" \/> in exponential form, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-704ce4ed76963f020a96a9532d2bc734_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#121;&#125;&#61;&#120;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"71\" style=\"vertical-align: -7px;\" \/><\/p>\n<p id=\"fs-id1167834229268\">We will use point plotting to graph the function. It will be easier to start with values of <em data-effect=\"italics\">y<\/em> and then get <em data-effect=\"italics\">x<\/em>.<\/p>\n<table class=\"unnumbered\" summary=\"This table has three columns and seven rows. The first row is a header row and it reads y, 1 over 3 to the y power equals x and (x, y). In the first column below y, we have negative 2, negative 1, 0, 1, 2, and 3. In the second column below 1 over 3 to the y power equals x we have 1 over 3 to the negative 2 power equals 3 squared which equals 9, 1 over 3 to the negative 1 power equals 3 to the first power which equals 3, 1 over 3 to the 0 power equals 1, 1 over 3 to the 1 power equals 1 over 3, 1 over 3 squared equals 1 over 9, and 1 over 3 cubed equals 1 over 27. In the third column below (x, y) we have (9, negative 2), (3, negative 1), (1, 0), (1 over 3, 1), (1 over 9, 2), and (1 over 27, 3).\" data-label=\"\">\n<thead>\n<tr>\n<th data-valign=\"middle\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-0af556714940c351c933bba8cf840796_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\" \/><\/th>\n<th data-valign=\"middle\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-d1f7c1d21318fa340ace2341a3b72907_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#121;&#125;&#61;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"67\" style=\"vertical-align: -7px;\" \/><\/th>\n<th data-valign=\"middle\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-aee61752ae042431152087f74b766103_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#44;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"39\" style=\"vertical-align: -4px;\" \/><\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td data-valign=\"middle\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-17c33e2329e29a62a80ad2b547b4753d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"21\" style=\"vertical-align: 0px;\" \/><\/td>\n<td data-valign=\"middle\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-42ef6be8e831e08b6ae45e05784e7343_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#45;&#50;&#125;&#61;&#123;&#51;&#125;&#94;&#123;&#50;&#125;&#61;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"116\" style=\"vertical-align: -7px;\" \/><\/td>\n<td data-valign=\"middle\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-12c3e71ed6b67af24d3ddfc620b524e6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#57;&#44;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"middle\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-7b34c01098c83fa602de54e9d74d63a9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"21\" style=\"vertical-align: -1px;\" \/><\/td>\n<td data-valign=\"middle\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-6b7d33e3619d94f8659387146890a001_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#45;&#49;&#125;&#61;&#123;&#51;&#125;&#94;&#123;&#49;&#125;&#61;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"116\" style=\"vertical-align: -7px;\" \/><\/td>\n<td data-valign=\"middle\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-0d66a71b8940b998e4f29f8cccda06d4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#44;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"middle\" data-align=\"left\">0<\/td>\n<td data-valign=\"middle\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-afcd06bc45a9c8daba099eda65a271bb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#48;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"64\" style=\"vertical-align: -7px;\" \/><\/td>\n<td data-valign=\"middle\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-d6c757b148db3a6707e7e1f199d80f9f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"middle\" data-align=\"left\">1<\/td>\n<td data-valign=\"middle\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-f658264b3b0d6196d9085de31d1f8a4a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#49;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"65\" style=\"vertical-align: -7px;\" \/><\/td>\n<td data-valign=\"middle\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-74ecd3905f6b094ff640d2e59831c91f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#44;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"40\" style=\"vertical-align: -7px;\" \/><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"middle\" data-align=\"left\">2<\/td>\n<td data-valign=\"middle\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-1a878cebd2842da17f07794e922ce65a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"65\" style=\"vertical-align: -7px;\" \/><\/td>\n<td data-valign=\"middle\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-0a93d49acdf0ac70a89de89981cf1495_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#57;&#125;&#44;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"40\" style=\"vertical-align: -7px;\" \/><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"middle\" data-align=\"left\">3<\/td>\n<td data-valign=\"middle\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-bbf4958620df1270a9cb07f78d5052f0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#51;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#55;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"72\" style=\"vertical-align: -7px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-b3e4f9217ba5edf17ef0a863fc98edcc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#55;&#125;&#44;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"47\" style=\"vertical-align: -7px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><span data-type=\"media\" id=\"fs-id1167831102974\" data-alt=\"This figure shows the logarithmic curve going through the points (1 over 3, 1), (1, 0), and (3, negative 1).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcbcintermediatealgebra\/wp-content\/uploads\/sites\/1021\/2020\/05\/CNX_IntAlg_Figure_10_03_006_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows the logarithmic curve going through the points (1 over 3, 1), (1, 0), and (3, negative 1).\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167834091580\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167834534505\">\n<div data-type=\"problem\" id=\"fs-id1167826940840\">\n<p>Graph: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-ef2b04a5a4cbf149a9e659f6f3b7e798_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#125;&#120;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"80\" style=\"vertical-align: -11px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835380353\">\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" data-alt=\"This figure shows the logarithmic curve going through the points (1 over 2, 1), (1, 0), and (2, negative 1).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcbcintermediatealgebra\/wp-content\/uploads\/sites\/1021\/2020\/05\/CNX_IntAlg_Figure_10_03_303_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows the logarithmic curve going through the points (1 over 2, 1), (1, 0), and (2, negative 1).\" \/><\/span><\/p>\n<p id=\"fs-id1167835340700\">\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167832059450\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167834429351\">\n<div data-type=\"problem\" id=\"fs-id1167835264420\">\n<p id=\"fs-id1167835218014\">Graph: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-1c91efb5fec7878308c06f27c38983fb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#125;&#120;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"80\" style=\"vertical-align: -11px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167831103870\" data-alt=\"This figure shows the logarithmic curve going through the points (1 over 4, 1), (1, 0), and (4, negative 1).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcbcintermediatealgebra\/wp-content\/uploads\/sites\/1021\/2020\/05\/CNX_IntAlg_Figure_10_03_304_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows the logarithmic curve going through the points (1 over 4, 1), (1, 0), and (4, negative 1).\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Now, let\u2019s look at the graphs <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-ada5cb914d058b00b45830e9ecaacf4f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#125;&#120;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#121;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#125;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"164\" style=\"vertical-align: -11px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-46b19d1c1b00b8928f1bf4a7ab7cbfec_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#125;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"76\" style=\"vertical-align: -11px;\" \/>, so we can identify some of the properties of logarithmic functions where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-ee823d38f20b11152c6360028dfe63e4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#60;&#97;&#60;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"79\" style=\"vertical-align: -1px;\" \/><\/p>\n<p id=\"fs-id1167834516390\">The graphs of all have the same basic shape. While this is the shape we expect from a logarithmic function where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-ee823d38f20b11152c6360028dfe63e4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#60;&#97;&#60;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"79\" style=\"vertical-align: -1px;\" \/><\/p>\n<p id=\"fs-id1167835595491\">We notice, that for each function again, the graph contains the points,<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-b2aedbffc1c0be6be268afc470cd42b2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"45\" style=\"vertical-align: -4px;\" \/><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-ca70a98fd69d9bc6c79a48c493db95b5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#44;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"46\" style=\"vertical-align: -4px;\" \/><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-6a1f5e5ad1e963280b4e34145679a907_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#97;&#125;&#44;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"63\" style=\"vertical-align: -7px;\" \/> This make sense for the same reasons we argued above.<\/p>\n<p id=\"fs-id1167835381440\">We notice the domain and range are also the same\u2014the domain is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-264878915584580308929a53513b9f1f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#92;&#105;&#110;&#102;&#116;&#121;&#32;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"47\" style=\"vertical-align: -4px;\" \/> and the range is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-5df92a2cc0a08c383e86c9bcbca032bf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#105;&#110;&#102;&#116;&#121;&#32;&#44;&#92;&#105;&#110;&#102;&#116;&#121;&#32;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"63\" style=\"vertical-align: -4px;\" \/> The <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-0af556714940c351c933bba8cf840796_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\" \/>-axis is again the vertical asymptote.<\/p>\n<p id=\"fs-id1167828396212\">We will summarize these properties in the chart below. Which also include when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-e81ef2f32f10882cbba77d80a157a26e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#62;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"46\" style=\"vertical-align: -1px;\" \/><\/p>\n<div data-type=\"note\" id=\"fs-id1167835233819\">\n<div data-type=\"title\">Properties of the Graph of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-88743ed111b206f89e0167ff135c8174_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#97;&#125;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"74\" style=\"vertical-align: -4px;\" \/><\/div>\n<table id=\"fs-id1167831882193\" class=\"unnumbered\" summary=\"Table has four columns. It shows that when a is greater than 1, the domain is 0 to infinity, the range is negative infinity to infinity, the x-intercept is 1, 0, there is no y intercept, the function contains a, 1 and , 1 over a, negative 1, the asymptote is the y axis 0, and the basic shape is increasing. It shows that when a is greater than 0 and less than 1, the domain is 0 to infinity, the range is negative infinity to infinity, the x-intercept is 1, 0, there is no y intercept, the function contains a, 1 and 1 over a, negative 1, the asymptote is the y axis, and the basic shape is decreasing.\" data-label=\"\">\n<thead>\n<tr>\n<th colspan=\"2\" data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">when<\/strong><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-eb7aa260ff031e44ade9124e9760983a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#62;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"41\" style=\"vertical-align: -1px;\" \/><\/th>\n<th colspan=\"2\" data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">when<\/strong><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-03771032443c255e636ebc73ff1819b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#60;&#97;&#60;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"74\" style=\"vertical-align: -1px;\" \/><\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Domain<\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-264878915584580308929a53513b9f1f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#92;&#105;&#110;&#102;&#116;&#121;&#32;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"47\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\">Domain<\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-264878915584580308929a53513b9f1f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#92;&#105;&#110;&#102;&#116;&#121;&#32;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"47\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Range<\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-90eaa2f0f55290692476556ec134a082_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#105;&#110;&#102;&#116;&#121;&#32;&#44;&#92;&#105;&#110;&#102;&#116;&#121;&#32;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"56\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\">Range<\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-90eaa2f0f55290692476556ec134a082_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#105;&#110;&#102;&#116;&#121;&#32;&#44;&#92;&#105;&#110;&#102;&#116;&#121;&#32;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"56\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/>-intercept<\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-d6c757b148db3a6707e7e1f199d80f9f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/>-intercept<\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-d6c757b148db3a6707e7e1f199d80f9f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-0af556714940c351c933bba8cf840796_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\" \/>-intercept<\/td>\n<td data-valign=\"top\" data-align=\"left\">none<\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-0af556714940c351c933bba8cf840796_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\" \/>-intercept<\/td>\n<td data-valign=\"top\" data-align=\"left\">None<\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Contains<\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-ca70a98fd69d9bc6c79a48c493db95b5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#44;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"46\" style=\"vertical-align: -4px;\" \/><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-5ef51d1178752973f3de098e0149e761_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#97;&#125;&#44;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"54\" style=\"vertical-align: -7px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\">Contains<\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-ca70a98fd69d9bc6c79a48c493db95b5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#44;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"46\" style=\"vertical-align: -4px;\" \/><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-5ef51d1178752973f3de098e0149e761_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#97;&#125;&#44;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"54\" style=\"vertical-align: -7px;\" \/><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Asymptote<\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-0af556714940c351c933bba8cf840796_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\" \/>-axis<\/td>\n<td data-valign=\"top\" data-align=\"left\">Asymptote<\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-0af556714940c351c933bba8cf840796_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\" \/>-axis<\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Basic shape<\/td>\n<td data-valign=\"top\" data-align=\"left\">increasing<\/td>\n<td data-valign=\"top\" data-align=\"left\">Basic shape<\/td>\n<td data-valign=\"top\" data-align=\"left\">Decreasing<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><span data-type=\"media\" data-alt=\"This figure shows that, for a greater than 1, the logarithmic curve going through the points (1 over a, negative 1), (1, 0), and (a, 1). This figure shows that, for a greater than 0 and less than 1, the logarithmic curve going through the points (a, 1), (1, 0), and (1 over a, negative 1).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcbcintermediatealgebra\/wp-content\/uploads\/sites\/1021\/2020\/05\/CNX_IntAlg_Figure_10_03_007_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows that, for a greater than 1, the logarithmic curve going through the points (1 over a, negative 1), (1, 0), and (a, 1). This figure shows that, for a greater than 0 and less than 1, the logarithmic curve going through the points (a, 1), (1, 0), and (1 over a, negative 1).\" \/><\/span><\/p>\n<\/div>\n<p id=\"fs-id1167835510177\">We talked earlier about how the logarithmic function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-06086e3c417416fef274d2f6241b01bf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#102;&#125;&#94;&#123;&#45;&#49;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#97;&#125;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"120\" style=\"vertical-align: -4px;\" \/> is the inverse of the exponential function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-a94bf83ed49b475db5990bf43c61d4c6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#97;&#125;&#94;&#123;&#120;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"83\" style=\"vertical-align: -4px;\" \/> The graphs in <a href=\"#CNX_IntAlg_Figure_10_03_008_img\" class=\"autogenerated-content\">(Figure)<\/a> show both the exponential (blue) and logarithmic (red) functions on the same graph for both <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-eb7aa260ff031e44ade9124e9760983a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#62;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"41\" style=\"vertical-align: -1px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-ee823d38f20b11152c6360028dfe63e4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#60;&#97;&#60;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"79\" style=\"vertical-align: -1px;\" \/><\/p>\n<div class=\"bc-figure figure\" id=\"CNX_IntAlg_Figure_10_03_008_img\"><span data-type=\"media\" id=\"fs-id1167832074212\" data-alt=\"This figure shows that, for a greater than 1, the logarithmic curve going through the points (1 over a, negative 1), (1, 0), and (a, 1). It also shows the exponential curve going through the points (1, 1 over a), (0, 1), and (1, a) along with the line y equals x. The logarithmic curve is a mirror image of the exponential curve across the y equals x line. This figure shows that, for a greater than 0 and less than 1, the logarithmic curve going through the points (a, 1), (1, 0), and (1 over a, negative 1). It also shows the exponential curve going through the points (negative 1, 1 over a), (0, 1), and (1, a) along with the line y equals x. The logarithmic curve is a mirror image of the exponential curve across the y equals x line.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcbcintermediatealgebra\/wp-content\/uploads\/sites\/1021\/2020\/05\/CNX_IntAlg_Figure_10_03_008_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows that, for a greater than 1, the logarithmic curve going through the points (1 over a, negative 1), (1, 0), and (a, 1). It also shows the exponential curve going through the points (1, 1 over a), (0, 1), and (1, a) along with the line y equals x. The logarithmic curve is a mirror image of the exponential curve across the y equals x line. This figure shows that, for a greater than 0 and less than 1, the logarithmic curve going through the points (a, 1), (1, 0), and (1 over a, negative 1). It also shows the exponential curve going through the points (negative 1, 1 over a), (0, 1), and (1, a) along with the line y equals x. The logarithmic curve is a mirror image of the exponential curve across the y equals x line.\" \/><\/span><\/div>\n<p id=\"fs-id1167835417624\">Notice how the graphs are reflections of each other through the line <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-e7685b887e3d125a1e0ead8be22eccc6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#120;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"47\" style=\"vertical-align: -4px;\" \/> We know this is true of inverse functions. Keeping a visual in your mind of these graphs will help you remember the domain and range of each function. Notice the <em data-effect=\"italics\">x<\/em>-axis is the horizontal asymptote for the exponential functions and the <em data-effect=\"italics\">y<\/em>-axis is the vertical asymptote for the logarithmic functions.<\/p>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167834473423\">\n<h3 data-type=\"title\">Solve Logarithmic Equations<\/h3>\n<p id=\"fs-id1167832015982\">When we talked about exponential functions, we introduced the number <em data-effect=\"italics\">e<\/em>. Just as <em data-effect=\"italics\">e<\/em> was a base for an exponential function, it can be used a base for logarithmic functions too. The logarithmic function with base <em data-effect=\"italics\">e<\/em> is called the <span data-type=\"term\">natural logarithmic function<\/span>. The function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-8918c8e8bbaa7c4d7335fa05cabfa451_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#101;&#125;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"101\" style=\"vertical-align: -4px;\" \/> is generally written <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-56e1c7d8eea20bd2ebeb56beb8c81bc0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#110;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"89\" style=\"vertical-align: -4px;\" \/> and we read it as \u201cel en of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-dffd8a2b2ed718a8a727c409b15d18e8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#46;&#39;&#39;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"23\" style=\"vertical-align: 0px;\" \/><\/p>\n<div data-type=\"note\" id=\"fs-id1167835351566\">\n<div data-type=\"title\">Natural Logarithmic Function<\/div>\n<p id=\"fs-id1167835234222\">The function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-56e1c7d8eea20bd2ebeb56beb8c81bc0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#110;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"89\" style=\"vertical-align: -4px;\" \/> is the <strong data-effect=\"bold\">natural logarithmic function<\/strong> with base <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-cdd204f9eb9bf059f6c66abbb4af16d5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#101;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"12\" style=\"vertical-align: -4px;\" \/> where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-68c3061ebd908643e1b359493c6a0a35_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#62;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"47\" style=\"vertical-align: 0px;\" \/><\/p>\n<div data-type=\"equation\" id=\"fs-id1167835318932\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-2538f084b4ef090b588044b988a4420b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#110;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#105;&#115;&#32;&#101;&#113;&#117;&#105;&#118;&#97;&#108;&#101;&#110;&#116;&#32;&#116;&#111;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#61;&#123;&#101;&#125;&#94;&#123;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"235\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/div>\n<p id=\"fs-id1167834448872\">When the base of the logarithm function is 10, we call it the <span data-type=\"term\">common logarithmic function<\/span> and the base is not shown. If the base <em data-effect=\"italics\">a<\/em> of a logarithm is not shown, we assume it is 10.<\/p>\n<div data-type=\"note\" id=\"fs-id1167831214279\">\n<div data-type=\"title\">Common Logarithmic Function<\/div>\n<p id=\"fs-id1167835311021\">The function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-7a38f69803eed430423e440fd4880f32_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"97\" style=\"vertical-align: -4px;\" \/> is the <strong data-effect=\"bold\">common logarithmic function<\/strong> with base<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-5450913cc453faf132dc56e5965ca797_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"17\" style=\"vertical-align: -1px;\" \/>, where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-68c3061ebd908643e1b359493c6a0a35_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#62;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"47\" style=\"vertical-align: 0px;\" \/><\/p>\n<div data-type=\"equation\" id=\"fs-id1167831227947\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-3cef176c6374bd4e34e9872b10c2b32f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#105;&#115;&#32;&#101;&#113;&#117;&#105;&#118;&#97;&#108;&#101;&#110;&#116;&#32;&#116;&#111;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#61;&#123;&#49;&#48;&#125;&#94;&#123;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"253\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/div>\n<p><span data-type=\"media\" id=\"fs-id1167826967294\" data-alt=\"It will be important for you to use your calculator to evaluate both common and natural logarithms. Find the log and ln keys on your calculator.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcbcintermediatealgebra\/wp-content\/uploads\/sites\/1021\/2020\/05\/CNX_IntAlg_Figure_10_03_009_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"It will be important for you to use your calculator to evaluate both common and natural logarithms. Find the log and ln keys on your calculator.\" \/><\/span><\/p>\n<p id=\"fs-id1167835416834\">To solve logarithmic equations, one strategy is to change the equation to exponential form and then solve the exponential equation as we did before. As we solve logarithmic equations, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-88743ed111b206f89e0167ff135c8174_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#97;&#125;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"74\" style=\"vertical-align: -4px;\" \/>, we need to remember that for the base <em data-effect=\"italics\">a<\/em>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-fee5d58cd9baf2f82bb3669852be05fa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#62;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"42\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-31658a6c5c3f18d97495ba25506408f2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#92;&#110;&#101;&#32;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"46\" style=\"vertical-align: -4px;\" \/> Also, the domain is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-68c3061ebd908643e1b359493c6a0a35_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#62;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"47\" style=\"vertical-align: 0px;\" \/> Just as with radical equations, we must check our solutions to eliminate any extraneous solutions.<\/p>\n<div data-type=\"example\" id=\"fs-id1167826996479\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167834473642\">\n<div data-type=\"problem\" id=\"fs-id1167835311985\">\n<p id=\"fs-id1167835380844\">Solve: <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-56a65c68f258164d6e5ce90046c18634_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#97;&#125;&#52;&#57;&#61;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"80\" style=\"vertical-align: -4px;\" \/> and <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-47dd42b97cd9c65344b0a4f1e2223517_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#110;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#61;&#51;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"65\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835181600\">\n<p id=\"fs-id1167835240986\"><span class=\"token\">\u24d0<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-9e4d130484f407830901246d5cca4a12_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#97;&#125;&#52;&#57;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#82;&#101;&#119;&#114;&#105;&#116;&#101;&#32;&#105;&#110;&#32;&#101;&#120;&#112;&#111;&#110;&#101;&#110;&#116;&#105;&#97;&#108;&#32;&#102;&#111;&#114;&#109;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#52;&#57;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#111;&#108;&#118;&#101;&#32;&#116;&#104;&#101;&#32;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#32;&#117;&#115;&#105;&#110;&#103;&#32;&#116;&#104;&#101;&#32;&#115;&#113;&#117;&#97;&#114;&#101;&#32;&#114;&#111;&#111;&#116;&#32;&#112;&#114;&#111;&#112;&#101;&#114;&#116;&#121;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#97;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&plusmn;&#55;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#104;&#101;&#32;&#98;&#97;&#115;&#101;&#32;&#99;&#97;&#110;&#110;&#111;&#116;&#32;&#98;&#101;&#32;&#110;&#101;&#103;&#97;&#116;&#105;&#118;&#101;&#44;&#32;&#115;&#111;&#32;&#119;&#101;&#32;&#101;&#108;&#105;&#109;&#105;&#110;&#97;&#116;&#101;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#97;&#61;&#45;&#55;&#46;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#56;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#97;&#61;&#55;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#111;&#118;&#101;&#114;&#108;&#105;&#110;&#101;&#123;&#41;&#97;&#61;&#45;&#55;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#104;&#101;&#99;&#107;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#97;&#61;&#55;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#97;&#125;&#52;&#57;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#55;&#125;&#52;&#57;&#38;&#32;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#63;&#125;&#123;&#61;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#55;&#125;&#94;&#123;&#50;&#125;&#38;&#32;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#63;&#125;&#123;&#61;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#52;&#57;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#52;&#57;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#52;&#57;&#10003;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"224\" width=\"560\" style=\"vertical-align: -95px;\" \/><\/p>\n<p id=\"fs-id1167835259254\"><span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-37b200b07d91e26da215aa28b063ceff_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#56;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#110;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#82;&#101;&#119;&#114;&#105;&#116;&#101;&#32;&#105;&#110;&#32;&#101;&#120;&#112;&#111;&#110;&#101;&#110;&#116;&#105;&#97;&#108;&#32;&#102;&#111;&#114;&#109;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#55;&#46;&#55;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#101;&#125;&#94;&#123;&#51;&#125;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#120;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#104;&#101;&#99;&#107;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#120;&#61;&#123;&#101;&#125;&#94;&#123;&#51;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#110;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#110;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#101;&#125;&#94;&#123;&#51;&#125;&#38;&#32;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#63;&#125;&#123;&#61;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#101;&#125;&#94;&#123;&#51;&#125;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#123;&#101;&#125;&#94;&#123;&#51;&#125;&#10003;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"129\" width=\"518\" style=\"vertical-align: -58px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167826997664\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835253878\">\n<div data-type=\"problem\" id=\"fs-id1167835595145\">\n<p id=\"fs-id1167834134740\">Solve: <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-35e2e8abc9822abba13cc206986d4fd0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#97;&#125;&#49;&#50;&#49;&#61;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"89\" style=\"vertical-align: -4px;\" \/> <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-153bb7592d4f1473ce7886bc75796f29_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#110;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#61;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"61\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835511251\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-2b15e1af4fec7b5f55f294ed901b7df8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#61;&#49;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"50\" style=\"vertical-align: -1px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-6e2300ab222cf7b01bd8a5b575628a16_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#123;&#101;&#125;&#94;&#123;&#55;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"49\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167834432020\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835236467\">\n<div data-type=\"problem\" id=\"fs-id1167835330555\">\n<p id=\"fs-id1167826781244\">Solve: <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-0c02c7a04d8c0f5dd85a5fdf1ad1e9f4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#97;&#125;&#54;&#52;&#61;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"81\" style=\"vertical-align: -4px;\" \/> <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-fbe2946afa1cdbd419dca732b1a9bc8c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#110;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#61;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"61\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831888043\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-e1735c51b99ed2469a7f2a6f728de75f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#61;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"42\" style=\"vertical-align: -1px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-b78fb8b1431a83f2263dd8e45c328f46_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#123;&#101;&#125;&#94;&#123;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"49\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"example\" id=\"fs-id1167831040648\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167835257318\">\n<div data-type=\"problem\" id=\"fs-id1167830697885\">\n<p id=\"fs-id1167831908789\">Solve: <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-7d88a1e3b7d938f11839f91693aabf5a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#50;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#120;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"129\" style=\"vertical-align: -4px;\" \/> and <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-b70b4d49b0e838dedd116758ad5e4328_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#110;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#101;&#125;&#94;&#123;&#50;&#120;&#125;&#61;&#52;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"79\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<p id=\"fs-id1167835173443\"><span class=\"token\">\u24d0<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-85e6505f5ecf66411e984c6b58eae643_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#50;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#120;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#82;&#101;&#119;&#114;&#105;&#116;&#101;&#32;&#105;&#110;&#32;&#101;&#120;&#112;&#111;&#110;&#101;&#110;&#116;&#105;&#97;&#108;&#32;&#102;&#111;&#114;&#109;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#50;&#125;&#94;&#123;&#52;&#125;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#51;&#120;&#45;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#105;&#109;&#112;&#108;&#105;&#102;&#121;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#49;&#54;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#51;&#120;&#45;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#111;&#108;&#118;&#101;&#32;&#116;&#104;&#101;&#32;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#50;&#49;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#51;&#120;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#55;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#120;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#104;&#101;&#99;&#107;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#120;&#61;&#55;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#50;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#120;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#50;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#92;&#99;&#100;&#111;&#116;&#32;&#55;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#38;&#32;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#63;&#125;&#123;&#61;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#50;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#38;&#32;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#63;&#125;&#123;&#61;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#50;&#125;&#94;&#123;&#52;&#125;&#38;&#32;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#63;&#125;&#123;&#61;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#49;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#49;&#54;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#49;&#54;&#10003;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"251\" width=\"552\" style=\"vertical-align: -119px;\" \/><\/p>\n<p id=\"fs-id1167831922589\"><span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-fc27df71a0da715b68d03cc8b6f001ff_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#108;&#110;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#101;&#125;&#94;&#123;&#50;&#120;&#125;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#82;&#101;&#119;&#114;&#105;&#116;&#101;&#32;&#105;&#110;&#32;&#101;&#120;&#112;&#111;&#110;&#101;&#110;&#116;&#105;&#97;&#108;&#32;&#102;&#111;&#114;&#109;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#101;&#125;&#94;&#123;&#52;&#125;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#123;&#101;&#125;&#94;&#123;&#50;&#120;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#105;&#110;&#99;&#101;&#32;&#116;&#104;&#101;&#32;&#98;&#97;&#115;&#101;&#115;&#32;&#97;&#114;&#101;&#32;&#116;&#104;&#101;&#32;&#115;&#97;&#109;&#101;&#32;&#116;&#104;&#101;&#32;&#101;&#120;&#112;&#111;&#110;&#101;&#110;&#116;&#115;&#32;&#97;&#114;&#101;&#32;&#101;&#113;&#117;&#97;&#108;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#52;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#50;&#120;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#111;&#108;&#118;&#101;&#32;&#116;&#104;&#101;&#32;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#50;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#120;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#104;&#101;&#99;&#107;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#120;&#61;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#108;&#110;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#101;&#125;&#94;&#123;&#50;&#120;&#125;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#108;&#110;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#101;&#125;&#94;&#123;&#50;&middot;&#50;&#125;&#38;&#32;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#63;&#125;&#123;&#61;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#108;&#110;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#101;&#125;&#94;&#123;&#52;&#125;&#38;&#32;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#63;&#125;&#123;&#61;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#101;&#125;&#94;&#123;&#52;&#125;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#123;&#101;&#125;&#94;&#123;&#52;&#125;&#10003;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"202\" width=\"577\" style=\"vertical-align: -93px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167834423068\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167834423072\">\n<div data-type=\"problem\" id=\"fs-id1167834556830\">\n<p id=\"fs-id1167834556832\">Solve: <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-8aed92d88b3fbf50ede4515e507c20fc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#50;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#120;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"129\" style=\"vertical-align: -4px;\" \/> <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-6eb067d1e14cd6a3675ba0fd13eb2a53_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#110;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#101;&#125;&#94;&#123;&#51;&#120;&#125;&#61;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"75\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834462946\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-acd0789444311f29aa58002088c8e0eb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#49;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"52\" style=\"vertical-align: -1px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-c657687cbbf5ea9a7545edb42190e592_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"42\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835421044\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835421048\">\n<div data-type=\"problem\" id=\"fs-id1167835342572\">\n<p id=\"fs-id1167835342574\">Solve: <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-913c0d4365c4fb3e99ce73e585b7ca1b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#51;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#120;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"129\" style=\"vertical-align: -4px;\" \/> <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-5211034d97c9755b34c8afa0a1b03c80_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#110;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#101;&#125;&#94;&#123;&#52;&#120;&#125;&#61;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"75\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-116dfacb4c2a32bdfb5ab9e852f29986_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"43\" style=\"vertical-align: 0px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-3330a01aa4d7d81947b71297d8623d3b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"42\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167831920506\">\n<h3 data-type=\"title\">Use Logarithmic Models in Applications<\/h3>\n<p id=\"fs-id1167834301111\">There are many applications that are modeled by logarithmic equations. We will first look at the logarithmic equation that gives the decibel (dB) level of sound. Decibels range from 0, which is barely audible to 160, which can rupture an eardrum. The <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-f4eafea8ecbb00d81b6d9726802f75e2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#49;&#48;&#125;&#94;&#123;&#45;&#49;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"42\" style=\"vertical-align: -1px;\" \/> in the formula represents the intensity of sound that is barely audible.<\/p>\n<div data-type=\"note\" id=\"fs-id1167827987443\">\n<div data-type=\"title\">Decibel Level of Sound<\/div>\n<p id=\"fs-id1167827987448\">The loudness level, <em data-effect=\"italics\">D<\/em>, measured in decibels, of a sound of intensity, <em data-effect=\"italics\">I<\/em>, measured in watts per square inch is<\/p>\n<div data-type=\"equation\" id=\"fs-id1167834191577\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-8e11245349956b5cd4abb23e836889a3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#68;&#61;&#49;&#48;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#73;&#125;&#123;&#123;&#49;&#48;&#125;&#94;&#123;&#45;&#49;&#50;&#125;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"139\" style=\"vertical-align: -8px;\" \/><\/div>\n<\/div>\n<div data-type=\"example\" id=\"fs-id1167834062530\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167834062533\">\n<div data-type=\"problem\" id=\"fs-id1167834062535\">\n<p id=\"fs-id1167832149826\">Extended exposure to noise that measures 85 dB can cause permanent damage to the inner ear which will result in hearing loss. What is the decibel level of music coming through ear phones with intensity <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-a2adce90e7fd78901b14540423deb0c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#49;&#48;&#125;&#94;&#123;&#45;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"35\" style=\"vertical-align: -1px;\" \/> watts per square inch?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834098991\">\n<table id=\"fs-id1167835359596\" class=\"unnumbered unstyled\" summary=\"We start with D equals 10 times log of the quantity I over 10 to the negative 12 power. We substitute in the intensity level I to obtain D equals 10 log of the quantity10 to the negative 2 power over 10 to the negative 12 power. We then simplify to obtain D equals 10 times log of 10 to the 10 power. Since log of 10 to the 10 equals 10, we have that D equals 10 times 10. Multiplying gives that D equals 100. Hence, the decibel level of music coming through earphones is 100 dB.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834183461\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcbcintermediatealgebra\/wp-content\/uploads\/sites\/1021\/2020\/05\/CNX_IntAlg_Figure_10_03_010a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Substitute in the intensity level, <em data-effect=\"italics\">I.<\/em><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167830961870\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcbcintermediatealgebra\/wp-content\/uploads\/sites\/1021\/2020\/05\/CNX_IntAlg_Figure_10_03_010b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832066044\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcbcintermediatealgebra\/wp-content\/uploads\/sites\/1021\/2020\/05\/CNX_IntAlg_Figure_10_03_010c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Since <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-2393791822afaa6d3b16a7dbfc5b478e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#123;&#49;&#48;&#125;&#94;&#123;&#49;&#48;&#125;&#61;&#49;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"101\" style=\"vertical-align: -3px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834094687\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcbcintermediatealgebra\/wp-content\/uploads\/sites\/1021\/2020\/05\/CNX_IntAlg_Figure_10_03_010d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Multiply.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835334481\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcbcintermediatealgebra\/wp-content\/uploads\/sites\/1021\/2020\/05\/CNX_IntAlg_Figure_10_03_010e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"left\">The decibel level of music coming through earphones is 100 dB.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167831106999\" class=\"try\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167835596124\">\n<p id=\"fs-id1167835329663\">What is the decibel level of one of the new quiet dishwashers with intensity <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-cba199993a1ce2b1fbd835dfd62f57f6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#49;&#48;&#125;&#94;&#123;&#45;&#55;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"35\" style=\"vertical-align: -1px;\" \/> watts per square inch?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834532484\">\n<p id=\"fs-id1167834532486\">The quiet dishwashers have a decibel level of 50 dB.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167826778780\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167832067744\">\n<div data-type=\"problem\" id=\"fs-id1167832067746\">\n<p id=\"fs-id1167832067748\">What is the decibel level heavy city traffic with intensity <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-eea810a09e75c2035d30b334ec50e8ef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#49;&#48;&#125;&#94;&#123;&#45;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"35\" style=\"vertical-align: -1px;\" \/> watts per square inch?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835377416\">\n<p id=\"fs-id1167835377418\">The decibel level of heavy traffic is 90 dB.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167835595015\">The magnitude <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-dae6bae3dcdac4629730754352c5e329_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/> of an earthquake is measured by a logarithmic scale called the Richter scale. The model is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-389a1214d545f476e84cfadd809f5dee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#73;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"77\" style=\"vertical-align: -4px;\" \/> where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-18b5e45cb4a1ee02e81b9a980f828db8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#73;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/> is the intensity of the shock wave. This model provides a way to measure <span data-type=\"term\" class=\"no-emphasis\">earthquake intensity<\/span>.<\/p>\n<div data-type=\"note\" id=\"fs-id1167826857422\">\n<div data-type=\"title\">Earthquake Intensity<\/div>\n<p id=\"fs-id1167835235048\">The magnitude <em data-effect=\"italics\">R<\/em> of an earthquake is measured by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-389a1214d545f476e84cfadd809f5dee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#73;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"77\" style=\"vertical-align: -4px;\" \/> where <em data-effect=\"italics\">I<\/em> is the intensity of its shock wave.<\/p>\n<\/div>\n<div data-type=\"example\" id=\"fs-id1167831893196\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167834537686\">\n<div data-type=\"problem\" id=\"fs-id1167834537688\">\n<p id=\"fs-id1167834537690\">In 1906, San Francisco experienced an intense earthquake with a magnitude of 7.8 on the Richter scale. Over 80% of the city was destroyed by the resulting fires. In 2014, Los Angeles experienced a moderate earthquake that measured 5.1 on the Richter scale and caused ?108 million dollars of damage. Compare the intensities of the two earthquakes.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167832066561\">\n<p id=\"fs-id1167832066563\">To compare the intensities, we first need to convert the magnitudes to intensities using the log formula. Then we will set up a ratio to compare the intensities.<\/p>\n<p id=\"fs-id1167835353005\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-f73d77255ba10b68f270837782344dfe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#111;&#110;&#118;&#101;&#114;&#116;&#32;&#116;&#104;&#101;&#32;&#109;&#97;&#103;&#110;&#105;&#116;&#117;&#100;&#101;&#115;&#32;&#116;&#111;&#32;&#105;&#110;&#116;&#101;&#110;&#115;&#105;&#116;&#105;&#101;&#115;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#54;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#82;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#73;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#57;&#48;&#54;&#32;&#101;&#97;&#114;&#116;&#104;&#113;&#117;&#97;&#107;&#101;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#55;&#46;&#56;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#73;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#111;&#110;&#118;&#101;&#114;&#116;&#32;&#116;&#111;&#32;&#101;&#120;&#112;&#111;&#110;&#101;&#110;&#116;&#105;&#97;&#108;&#32;&#102;&#111;&#114;&#109;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#56;&#54;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#73;&#61;&#123;&#49;&#48;&#125;&#94;&#123;&#55;&#46;&#56;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#48;&#49;&#52;&#32;&#101;&#97;&#114;&#116;&#104;&#113;&#117;&#97;&#107;&#101;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#53;&#46;&#49;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#73;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#111;&#110;&#118;&#101;&#114;&#116;&#32;&#116;&#111;&#32;&#101;&#120;&#112;&#111;&#110;&#101;&#110;&#116;&#105;&#97;&#108;&#32;&#102;&#111;&#114;&#109;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#57;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#73;&#61;&#123;&#49;&#48;&#125;&#94;&#123;&#53;&#46;&#49;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#70;&#111;&#114;&#109;&#32;&#97;&#32;&#114;&#97;&#116;&#105;&#111;&#32;&#111;&#102;&#32;&#116;&#104;&#101;&#32;&#105;&#110;&#116;&#101;&#110;&#115;&#105;&#116;&#105;&#101;&#115;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#73;&#110;&#116;&#101;&#110;&#115;&#105;&#116;&#121;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#102;&#111;&#114;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#49;&#57;&#48;&#54;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#73;&#110;&#116;&#101;&#110;&#115;&#105;&#116;&#121;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#102;&#111;&#114;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#50;&#48;&#49;&#52;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#117;&#98;&#115;&#116;&#105;&#116;&#117;&#116;&#101;&#32;&#105;&#110;&#32;&#116;&#104;&#101;&#32;&#118;&#97;&#108;&#117;&#101;&#115;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#49;&#48;&#125;&#94;&#123;&#55;&#46;&#56;&#125;&#125;&#123;&#123;&#49;&#48;&#125;&#94;&#123;&#53;&#46;&#49;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#68;&#105;&#118;&#105;&#100;&#101;&#32;&#98;&#121;&#32;&#115;&#117;&#98;&#116;&#114;&#97;&#99;&#116;&#105;&#110;&#103;&#32;&#116;&#104;&#101;&#32;&#101;&#120;&#112;&#111;&#110;&#101;&#110;&#116;&#115;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#49;&#48;&#125;&#94;&#123;&#50;&#46;&#55;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#69;&#118;&#97;&#108;&#117;&#97;&#116;&#101;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#53;&#48;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#104;&#101;&#32;&#105;&#110;&#116;&#101;&#110;&#115;&#105;&#116;&#121;&#32;&#111;&#102;&#32;&#116;&#104;&#101;&#32;&#49;&#57;&#48;&#54;&#32;&#101;&#97;&#114;&#116;&#104;&#113;&#117;&#97;&#107;&#101;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#119;&#97;&#115;&#32;&#97;&#98;&#111;&#117;&#116;&#32;&#53;&#48;&#49;&#32;&#116;&#105;&#109;&#101;&#115;&#32;&#116;&#104;&#101;&#32;&#105;&#110;&#116;&#101;&#110;&#115;&#105;&#116;&#121;&#32;&#111;&#102;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#32;&#50;&#48;&#49;&#52;&#32;&#101;&#97;&#114;&#116;&#104;&#113;&#117;&#97;&#107;&#101;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"308\" width=\"640\" style=\"vertical-align: -149px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167827943038\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167827943041\">\n<div data-type=\"problem\" id=\"fs-id1167831826460\">\n<p id=\"fs-id1167831826462\">In 1906, San Francisco experienced an intense earthquake with a magnitude of 7.8 on the Richter scale. In 1989, the Loma Prieta earthquake also affected the San Francisco area, and measured 6.9 on the Richter scale. Compare the intensities of the two earthquakes.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835229713\">\n<p id=\"fs-id1167835229715\">The intensity of the 1906 earthquake was about 8 times the intensity of the 1989 earthquake.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835345396\" class=\"try\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167835217831\">\n<p id=\"fs-id1167835217833\">In 2014, Chile experienced an intense earthquake with a magnitude of 8.2 on the Richter scale. In 2014, Los Angeles also experienced an earthquake which measured 5.1 on the Richter scale. Compare the intensities of the two earthquakes.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834131115\">\n<p id=\"fs-id1167831922298\">The intensity of the earthquake in Chile was about 1,259 times the intensity of the earthquake in Los Angeles.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167834065272\" class=\"media-2\">\n<p id=\"fs-id1167834065275\">Access these online resources for additional instruction and practice with evaluating and graphing logarithmic functions.<\/p>\n<ul id=\"fs-id1167835329751\" data-display=\"block\">\n<li><a href=\"https:\/\/openstax.org\/l\/37logasexponent\">Re-writing logarithmic equations in exponential form<\/a><\/li>\n<li><a href=\"https:\/\/openstax.org\/l\/37Simplifylog\">Simplifying Logarithmic Expressions<\/a><\/li>\n<li><a href=\"https:\/\/openstax.org\/l\/37Graphlog\">Graphing logarithmic functions<\/a><\/li>\n<li><a href=\"https:\/\/openstax.org\/l\/37Finddecibel\">Using logarithms to calculate decibel levels<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167832076686\">\n<h3 data-type=\"title\">Key Concepts<\/h3>\n<ul id=\"fs-id1167826827794\" data-bullet-style=\"bullet\">\n<li><strong data-effect=\"bold\">Properties of the Graph of<\/strong><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-52642b6a988dcf8bf2c2419306159679_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#97;&#125;&#120;&#58;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"83\" style=\"vertical-align: -4px;\" \/>\n<div data-type=\"newline\"><\/div>\n<table id=\"fs-id1167835329636\" class=\"unnumbered\" summary=\"Table has four columns. It shows that when a is greater than 1, the domain is 0 to infinity, the range is negative infinity to infinity, the x-intercept is 1, 0, there is no y intercept, the function contains a, 1 and , 1 over a, negative 1, the asymptote is the y axis 0, and the basic shape is increasing. It shows that when a is greater than 0 and less than 1, the domain is 0 to infinity, the range is negative infinity to infinity, the x-intercept is 1, 0, there is no y intercept, the function contains a, 1 and 1 over a, negative 1, the asymptote is the y axis, and the basic shape is decreasing.\" data-label=\"\">\n<thead>\n<tr>\n<th colspan=\"2\" data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">when<\/strong><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-eb7aa260ff031e44ade9124e9760983a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#62;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"41\" style=\"vertical-align: -1px;\" \/><\/th>\n<th colspan=\"2\" data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">when<\/strong><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-03771032443c255e636ebc73ff1819b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#60;&#97;&#60;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"74\" style=\"vertical-align: -1px;\" \/><\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Domain<\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-264878915584580308929a53513b9f1f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#92;&#105;&#110;&#102;&#116;&#121;&#32;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"47\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\">Domain<\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-264878915584580308929a53513b9f1f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#92;&#105;&#110;&#102;&#116;&#121;&#32;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"47\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Range<\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-90eaa2f0f55290692476556ec134a082_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#105;&#110;&#102;&#116;&#121;&#32;&#44;&#92;&#105;&#110;&#102;&#116;&#121;&#32;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"56\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\">Range<\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-90eaa2f0f55290692476556ec134a082_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#105;&#110;&#102;&#116;&#121;&#32;&#44;&#92;&#105;&#110;&#102;&#116;&#121;&#32;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"56\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\"><em data-effect=\"italics\">x<\/em>-intercept<\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-d6c757b148db3a6707e7e1f199d80f9f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><em data-effect=\"italics\">x<\/em>-intercept<\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-d6c757b148db3a6707e7e1f199d80f9f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\"><em data-effect=\"italics\">y<\/em>-intercept<\/td>\n<td data-valign=\"top\" data-align=\"left\">none<\/td>\n<td data-valign=\"top\" data-align=\"left\"><em data-effect=\"italics\">y<\/em>-intercept<\/td>\n<td data-valign=\"top\" data-align=\"left\">none<\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Contains<\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-ca70a98fd69d9bc6c79a48c493db95b5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#44;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"46\" style=\"vertical-align: -4px;\" \/><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-5ef51d1178752973f3de098e0149e761_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#97;&#125;&#44;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"54\" style=\"vertical-align: -7px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\">Contains<\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-ca70a98fd69d9bc6c79a48c493db95b5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#44;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"46\" style=\"vertical-align: -4px;\" \/><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-5ef51d1178752973f3de098e0149e761_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#97;&#125;&#44;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"54\" style=\"vertical-align: -7px;\" \/><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Asymptote<\/td>\n<td data-valign=\"top\" data-align=\"left\"><em data-effect=\"italics\">y<\/em>-axis<\/td>\n<td data-valign=\"top\" data-align=\"left\">Asymptote<\/td>\n<td data-valign=\"top\" data-align=\"left\"><em data-effect=\"italics\">y<\/em>-axis<\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Basic shape<\/td>\n<td data-valign=\"top\" data-align=\"left\">increasing<\/td>\n<td data-valign=\"top\" data-align=\"left\">Basic shape<\/td>\n<td data-valign=\"top\" data-align=\"left\">decreasing<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167835380217\" data-alt=\"This figure shows that, for a greater than 1, the logarithmic curve going through the points (1 over a, negative 1), (1, 0), and (a, 1). This figure shows that, for a greater than 0 and less than 1, the logarithmic curve going through the points (a, 1), (1, 0), and (1 over a, negative 1).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcbcintermediatealgebra\/wp-content\/uploads\/sites\/1021\/2020\/05\/CNX_IntAlg_Figure_10_03_012_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows that, for a greater than 1, the logarithmic curve going through the points (1 over a, negative 1), (1, 0), and (a, 1). This figure shows that, for a greater than 0 and less than 1, the logarithmic curve going through the points (a, 1), (1, 0), and (1 over a, negative 1).\" \/><\/span><\/li>\n<li><strong data-effect=\"bold\">Decibel Level of Sound:<\/strong> The loudness level, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-4b9ef1bbd23fd1b198de883813285620_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#68;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"15\" style=\"vertical-align: 0px;\" \/>, measured in decibels, of a sound of intensity, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-18b5e45cb4a1ee02e81b9a980f828db8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#73;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/>, measured in watts per square inch is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-2c9ea93bc2b32c54a1b7d2bd37a09eff_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#68;&#61;&#49;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#73;&#125;&#123;&#123;&#49;&#48;&#125;&#94;&#123;&#45;&#49;&#50;&#125;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"145\" style=\"vertical-align: -8px;\" \/><\/li>\n<li><strong data-effect=\"bold\">Earthquake Intensity:<\/strong> The magnitude <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-dae6bae3dcdac4629730754352c5e329_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/> of an earthquake is measured by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-389a1214d545f476e84cfadd809f5dee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#73;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"77\" style=\"vertical-align: -4px;\" \/> where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-18b5e45cb4a1ee02e81b9a980f828db8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#73;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/> is the intensity of its shock wave.<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167826874229\">\n<div class=\"practice-perfect\" data-depth=\"2\">\n<h4 data-type=\"title\">Practice Makes Perfect<\/h4>\n<p id=\"fs-id1167830757469\"><strong data-effect=\"bold\">Convert Between Exponential and Logarithmic Form<\/strong><\/p>\n<p>In the following exercises, convert from exponential to logarithmic form.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167835610117\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835610119\">\n<p id=\"fs-id1167835610121\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-cfeee31fdf97d95856ac8567ab9a4036_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#52;&#125;&#94;&#123;&#50;&#125;&#61;&#49;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"58\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835511476\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835511478\">\n<p id=\"fs-id1167835614916\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-edcc26a7d893b401f1c0e1da3ff46252_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#50;&#125;&#94;&#123;&#53;&#125;&#61;&#51;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"57\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834402936\">\n<p id=\"fs-id1167834402938\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-c2822ba060f6ad82c7f486937b67d0b7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#50;&#125;&#51;&#50;&#61;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"80\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835378536\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835351753\">\n<p id=\"fs-id1167835351755\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-2f52aca3b2dff22e12c39218c34c6fc0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#51;&#125;&#94;&#123;&#51;&#125;&#61;&#50;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"58\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826828342\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167826828344\">\n<p id=\"fs-id1167826828346\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-b61902d2432e0c783be3b45dd1ae9f9b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#53;&#125;&#94;&#123;&#51;&#125;&#61;&#49;&#50;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"66\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834064488\">\n<p id=\"fs-id1167834064490\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-c2d32dfb54c219d9ef23947d2d08c44c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#53;&#125;&#49;&#50;&#53;&#61;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"90\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167830698573\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167830698575\">\n<p id=\"fs-id1167835376182\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-49dac1b2f889a6968cd7150276cdce03_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#49;&#48;&#125;&#94;&#123;&#51;&#125;&#61;&#49;&#48;&#48;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"84\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826781041\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167826781043\">\n<p id=\"fs-id1167826781046\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-1e85bc74499c08265194754f59f7dc46_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#49;&#48;&#125;&#94;&#123;&#45;&#50;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#49;&#48;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"82\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834186366\">\n<p id=\"fs-id1167834186368\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-74e16a767a9d0e126d66619f749bf7e2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#49;&#48;&#48;&#125;&#61;&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"93\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831970028\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167831970030\">\n<p id=\"fs-id1167834534486\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-aca2fd0920dae733f4763c1cf0c8d9b8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#125;&#61;&#92;&#115;&#113;&#114;&#116;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"68\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831872216\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167831872218\">\n<p id=\"fs-id1167835511349\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-1d250d5e14c0a45501ef4eb0ba68c9a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#125;&#61;&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"69\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834429528\">\n<p id=\"fs-id1167834429530\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-020dca50cef0cf4caa7543494bd7bf7a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#120;&#125;&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#54;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"88\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832043157\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835167487\">\n<p id=\"fs-id1167835167489\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-b87a94f59d65f89b0fd9cb3e63578fa8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#51;&#50;&#125;&#94;&#123;&#120;&#125;&#61;&#92;&#115;&#113;&#114;&#116;&#91;&#52;&#93;&#123;&#51;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"84\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831081691\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834058866\">\n<p id=\"fs-id1167834058868\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-29759a15a8a734de8bd384d6490c9627_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#49;&#55;&#125;&#94;&#123;&#120;&#125;&#61;&#92;&#115;&#113;&#114;&#116;&#91;&#53;&#93;&#123;&#49;&#55;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"83\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835356149\">\n<p id=\"fs-id1167830914966\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-53a700bb70198a3fb43345904de9d9d1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#49;&#55;&#125;&#92;&#115;&#113;&#114;&#116;&#91;&#53;&#93;&#123;&#49;&#55;&#125;&#61;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"104\" style=\"vertical-align: -5px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835240982\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167826995298\">\n<p id=\"fs-id1167826995300\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-5a65d9345b02bda0360e101a6ff9085f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#49;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"72\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835340501\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834555158\">\n<p id=\"fs-id1167834555160\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-54da27e214c71bfe8d4a5fe4c347ae94_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#52;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#56;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"72\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835283043\">\n<p id=\"fs-id1167835283045\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-f43d21d448aaa2ca5a27d3b686770036_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#56;&#49;&#125;&#61;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"84\" style=\"vertical-align: -11px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167828411027\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835311339\">\n<p id=\"fs-id1167835311341\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-05c41696766926e7c4eec5204c947892_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#51;&#125;&#94;&#123;&#45;&#50;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"60\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167830865339\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167830865341\">\n<p id=\"fs-id1167835511733\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-a7618f5553cebbec8f593651d1847492_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#52;&#125;&#94;&#123;&#45;&#51;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#54;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"67\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835305342\">\n<p id=\"fs-id1167835305344\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-3226da52316137a0f9190ccc5b84dbe2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#52;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#54;&#52;&#125;&#61;&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"95\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826799078\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167826799080\">\n<p id=\"fs-id1167826799082\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-a6b6db857a9f14e11aace26d45e5748d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#101;&#125;&#94;&#123;&#120;&#125;&#61;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"49\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826997372\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834289504\">\n<p id=\"fs-id1167834289506\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-e133c51c340c67500d8f7619fa8c3a1e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#101;&#125;&#94;&#123;&#51;&#125;&#61;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"50\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835374439\">\n<p id=\"fs-id1167834403039\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-88bd34e9164854fe7a4254c3a964dc45_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#110;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#61;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"61\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167831031378\">In the following exercises, convert each logarithmic equation to exponential form.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167831031381\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167831031383\">\n<p id=\"fs-id1167835332820\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-8342e9672cc61c19496d1524b537cb8a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#52;&#125;&#54;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"81\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832152944\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834402694\">\n<p id=\"fs-id1167834402696\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-191805968255ab2ea27ac4a11fd22e3d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#50;&#125;&#54;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"81\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835302082\">\n<p id=\"fs-id1167835302084\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-3978989d702c5579cff8c6dc3f46dd10_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#52;&#61;&#123;&#50;&#125;&#94;&#123;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"57\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834556107\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834556109\">\n<p id=\"fs-id1167834556111\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-5b52d889d4a2081451d04cc4d568a513_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#120;&#125;&#56;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"81\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835351507\" class=\"material-set-2\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167831076443\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-8f465dd3d71d42a9fb915035ebaabc30_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#120;&#125;&#51;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"81\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835511183\">\n<p id=\"fs-id1167835511185\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-846ff6e0f98bf328c9967043ca009cef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#50;&#61;&#123;&#120;&#125;&#94;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"58\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835226504\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167832195745\">\n<p id=\"fs-id1167832195747\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-c9af345e19f47597d21da92d468a1528_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#49;&#50;&#125;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"78\" style=\"vertical-align: -5px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835510915\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835510918\">\n<p id=\"fs-id1167826994101\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-b182217d5df1d32d03962521bcd816ee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#55;&#125;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"71\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835331715\">\n<p id=\"fs-id1167835331717\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-9ac739d48d4d438d9884e0478e67f50e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#61;&#123;&#55;&#125;&#94;&#123;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"48\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835267509\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167826778870\">\n<p id=\"fs-id1167826778873\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-c86fde0e473992e18a7346975a7d5739_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#51;&#125;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"71\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835336419\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835336421\">\n<p id=\"fs-id1167835336423\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-3763dfdfbac18da9a19818963ce87de1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#57;&#125;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"71\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834246582\">\n<p id=\"fs-id1167835614859\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-e9e7ee5ab073ce242079c4dcef5122a8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#61;&#123;&#57;&#125;&#94;&#123;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"48\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167828447061\">\n<p id=\"fs-id1167831958004\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-7c075a906c07e424fd3da74b15d59249_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#52;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#49;&#48;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#49;&#48;&#44;&#48;&#48;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"123\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832212044\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835243983\">\n<p id=\"fs-id1167835243985\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-b1b9cc7de28398ab68ea221b71cb0343_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#49;&#48;&#125;&#49;&#44;&#48;&#48;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"113\" style=\"vertical-align: -5px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834195105\">\n<p id=\"fs-id1167834195107\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-3dae669eb7b7c35e190406789ce12780_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#44;&#48;&#48;&#48;&#61;&#123;&#49;&#48;&#125;&#94;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"91\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834300253\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834300255\">\n<p id=\"fs-id1167835373839\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-44c14513cf8281a414af2faf6afc65ae_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#101;&#125;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"72\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832058780\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167826983132\">\n<p id=\"fs-id1167826983134\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-b7fc21c65e07073eacc6218a00785c35_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#101;&#125;&#52;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"82\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835376396\">\n<p id=\"fs-id1167835376398\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-e5efaf70c8364ab5d85ffaa756429361_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#51;&#61;&#123;&#101;&#125;&#94;&#123;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"57\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167834058960\"><strong data-effect=\"bold\">Evaluate Logarithmic Functions<\/strong><\/p>\n<p id=\"fs-id1167830836813\">In the following exercises, find the value of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/> in each logarithmic equation.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167832123885\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167832123887\">\n<p id=\"fs-id1167832123889\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-e02362f114c4823c68571910bee724d7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#120;&#125;&#52;&#57;&#61;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"81\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834063090\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834063092\">\n<p id=\"fs-id1167834063094\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-7e69d3d20ad7c5232ea9dcf5b99ebf30_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#120;&#125;&#49;&#50;&#49;&#61;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"89\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834098316\">\n<p id=\"fs-id1167834098318\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-afcb2f5e8d3025d4ad3e54c6c8942e1b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#49;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"51\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832058679\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167831846682\">\n<p id=\"fs-id1167831846684\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-0648bd6f86e9988b2394f4f7b88c274f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#120;&#125;&#50;&#55;&#61;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"82\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826799348\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167831923461\">\n<p id=\"fs-id1167831923463\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-569e06842e019f3864c58de1720b78f5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#120;&#125;&#54;&#52;&#61;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"82\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834506100\">\n<p id=\"fs-id1167834506103\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-2145acc2878ed61214887e120f2485b7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"43\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835350317\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835350319\">\n<p id=\"fs-id1167834395147\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-f0bb4525d94efc98500ed9b26e65c96b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#51;&#125;&#120;&#61;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"73\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835375306\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835375308\">\n<p id=\"fs-id1167835358824\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-8617f5a993d41603738d8873d1e55208_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#53;&#125;&#120;&#61;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"73\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834462750\">\n<p id=\"fs-id1167835365659\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-bd200032faab729d3376085fad9115bc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#49;&#50;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"60\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835236219\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835236222\">\n<p id=\"fs-id1167835236224\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-08ebc23ef6fce07e9acfb363a842284d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#50;&#125;&#120;&#61;&#45;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"87\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167830914861\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835468575\">\n<p id=\"fs-id1167835468577\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-d22178dc8e39850ab393c12a9bacd07b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#51;&#125;&#120;&#61;&#45;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"86\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834246985\">\n<p id=\"fs-id1167834246987\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-7d3fb3d5b57e3c2b733967ae6e3d1035_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#52;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"57\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835258321\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835258323\">\n<p id=\"fs-id1167835258325\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-01f477bfb74bd888998b8bbe7c4595d6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#49;&#54;&#125;&#61;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"85\" style=\"vertical-align: -11px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835420083\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835420086\">\n<p id=\"fs-id1167835351531\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-e250b64d2adb1616436149a5d27de99d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#57;&#125;&#61;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"78\" style=\"vertical-align: -11px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834505061\">\n<p id=\"fs-id1167834505063\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-c657687cbbf5ea9a7545edb42190e592_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"42\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831882568\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167831882570\">\n<p id=\"fs-id1167831066105\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-e32abedce55ad02bbb34570726b21bb0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#125;&#54;&#52;&#61;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"85\" style=\"vertical-align: -11px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831115372\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835339774\">\n<p id=\"fs-id1167835339776\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-c529fa7bdddfe37d3d80c9116bc828eb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#57;&#125;&#125;&#56;&#49;&#61;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"85\" style=\"vertical-align: -11px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167830858557\">\n<p id=\"fs-id1167830858559\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-01f282abd343bbe6b83c45e54b86c6ed_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"56\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167834252266\">In the following exercises, find the exact value of each logarithm without using a calculator.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167831881072\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167831881075\">\n<p id=\"fs-id1167831881077\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-1924a3631c07c74346183d4b8574335a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#55;&#125;&#52;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"48\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834184863\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834184865\">\n<p id=\"fs-id1167834184868\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-a234670719d0eaf7c3524511a6d39bcf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#54;&#125;&#51;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"48\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835376014\">\n<p id=\"fs-id1167831823260\">2<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835356037\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835356039\">\n<p id=\"fs-id1167835356042\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-258f26228b24d50089cd21ee63b0e04a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#52;&#125;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835339780\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835339782\">\n<p id=\"fs-id1167835339784\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-3896fb87dc87e3b1a0f5c550635e6e2e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#53;&#125;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<p id=\"fs-id1167835339616\">0<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835200134\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835200136\">\n<p id=\"fs-id1167835200139\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-5c2bcd49b8c2264b5f135b0b43248be9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#49;&#54;&#125;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"46\" style=\"vertical-align: -5px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834556258\" class=\"material-set-2\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167834556262\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-b3f0e1b2d38ca64ed9140b72fde8210e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#50;&#55;&#125;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"46\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834228002\">\n<p id=\"fs-id1167834228004\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-34f0e870957984f6c69249b8cf4f5813_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826779062\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167826779064\">\n<p id=\"fs-id1167826779066\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-975005563ace1e4b1c3bdcf95f3ad033_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#125;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"41\" style=\"vertical-align: -11px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834536087\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834536089\">\n<p id=\"fs-id1167834193412\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-c6ba2425f2c970bc2aec02afb2c10033_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#125;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"42\" style=\"vertical-align: -11px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826938274\">\n<p id=\"fs-id1167826938277\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-17c33e2329e29a62a80ad2b547b4753d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"21\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835513008\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167831871914\">\n<p id=\"fs-id1167831871917\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-155184c1316abdbe9d3c022bffd42432_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#50;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#49;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"46\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832041687\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167832041689\">\n<p id=\"fs-id1167832041691\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-ac143e86a14f323d4e835fb8c8efe57b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#51;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#55;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"46\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835596456\">\n<p id=\"fs-id1167835596458\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-470cb162cf92c55d139f4f69216225e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"22\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831969720\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167828421347\">\n<p id=\"fs-id1167828421349\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-ef00ea12514af65850c011400b2156f5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#52;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#49;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"46\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834195940\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834195942\">\n<p id=\"fs-id1167834195944\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-5e96d58f8104e857ea7616a0f80c2d5c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#57;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#56;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"46\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826864570\">\n<p id=\"fs-id1167826864572\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-17c33e2329e29a62a80ad2b547b4753d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"21\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167832126086\"><strong data-effect=\"bold\">Graph Logarithmic Functions<\/strong><\/p>\n<p id=\"fs-id1167835529367\">In the following exercises, graph each logarithmic function.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167835529370\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835257995\">\n<p id=\"fs-id1167835257997\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-38aafeef0a642048ff912be800a92b59_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#50;&#125;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"73\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835377701\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835377703\">\n<p id=\"fs-id1167832066676\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-1483b36c2b515f34acd4e94cb99dde31_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#52;&#125;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"73\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835381928\"><span data-type=\"media\" id=\"fs-id1167835381931\" data-alt=\"This figure shows the logarithmic curve going through the points (1 over 4, negative 1), (1, 0), and (4, 1).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcbcintermediatealgebra\/wp-content\/uploads\/sites\/1021\/2020\/05\/CNX_IntAlg_Figure_10_03_306_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows the logarithmic curve going through the points (1 over 4, negative 1), (1, 0), and (4, 1).\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826802368\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167831880182\">\n<p id=\"fs-id1167831880184\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-4e88abf82a4551dbb548fa3b133cfbd7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#54;&#125;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"73\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826997598\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167826997601\">\n<p id=\"fs-id1167834324636\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-aa3f4b531e7a2f8b6bbc0a9970088cfc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#55;&#125;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"73\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835317768\"><span data-type=\"media\" id=\"fs-id1167835317771\" data-alt=\"This figure shows that the logarithmic curve going through the points (1 over 7, negative 1), (1, 0), and (7, 1).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcbcintermediatealgebra\/wp-content\/uploads\/sites\/1021\/2020\/05\/CNX_IntAlg_Figure_10_03_308_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows that the logarithmic curve going through the points (1 over 7, negative 1), (1, 0), and (7, 1).\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831920466\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167831920468\">\n<p id=\"fs-id1167831920470\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-95b5f7a5021ee7cab305567458899052_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#49;&#46;&#53;&#125;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"84\" style=\"vertical-align: -5px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826864301\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167826864303\">\n<p id=\"fs-id1167826864305\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-7f51283f71c943231e6a793d66022fab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#50;&#46;&#53;&#125;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"84\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835284763\"><span data-type=\"media\" id=\"fs-id1167831881934\" data-alt=\"This figure shows the logarithmic curve going through the points (2 over 5, negative 1), (1, 0), and (2.5, 1).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcbcintermediatealgebra\/wp-content\/uploads\/sites\/1021\/2020\/05\/CNX_IntAlg_Figure_10_03_310_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows the logarithmic curve going through the points (2 over 5, negative 1), (1, 0), and (2.5, 1).\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834120868\" class=\"material-set-2\">\n<div data-type=\"problem\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-d545442297fc0ab3aa42cf31f90b8699_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#125;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"76\" style=\"vertical-align: -11px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835229539\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-1e3ee6fbf0c50b73d8ee039bf665aaf1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#53;&#125;&#125;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"76\" style=\"vertical-align: -11px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835287695\"><span data-type=\"media\" id=\"fs-id1167835287698\" data-alt=\"This figure shows the logarithmic curve going through the points (1 over 5, 1), (1, 0), and (5, negative 1).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcbcintermediatealgebra\/wp-content\/uploads\/sites\/1021\/2020\/05\/CNX_IntAlg_Figure_10_03_312_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows the logarithmic curve going through the points (1 over 5, 1), (1, 0), and (5, negative 1).\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835287628\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835287630\">\n<p id=\"fs-id1167835287632\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-478a566be4b7d306ab4352efa50d434f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#48;&#46;&#52;&#125;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"84\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834395367\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835335823\">\n<p id=\"fs-id1167835335825\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-5537a179a07d3dc48c275459d811dfc1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#48;&#46;&#54;&#125;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"84\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834536075\"><span data-type=\"media\" id=\"fs-id1167826825347\" data-alt=\"This figure shows the logarithmic curve going through the points (3 over 5, 1), (1, 0), and (5 over 3, negative 1).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcbcintermediatealgebra\/wp-content\/uploads\/sites\/1021\/2020\/05\/CNX_IntAlg_Figure_10_03_314_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows the logarithmic curve going through the points (3 over 5, 1), (1, 0), and (5 over 3, negative 1).\" \/><\/span><\/div>\n<\/div>\n<p id=\"fs-id1167831908264\"><strong data-effect=\"bold\">Solve Logarithmic Equations<\/strong><\/p>\n<p id=\"fs-id1167826998243\">In the following exercises, solve each logarithmic equation.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167826998246\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835375347\">\n<p id=\"fs-id1167835375350\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-3146368043952285850dc9ff96905ca1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#97;&#125;&#49;&#54;&#61;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"80\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835337605\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835337608\">\n<p id=\"fs-id1167835337610\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-70d452acf213438f6b7fab4b3a51f1ca_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#97;&#125;&#56;&#49;&#61;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"80\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834422749\">\n<p id=\"fs-id1167832066968\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-d15b5cf863ec5708ebc20c403ef571b5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#61;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"42\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835354040\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835354043\">\n<p id=\"fs-id1167835419126\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-611f820c53fad6ea365d8acfa2463771_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#97;&#125;&#56;&#61;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"72\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835518006\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167828240489\">\n<p id=\"fs-id1167828240491\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-56bd043aed1c5376fc82644c76b0591f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#97;&#125;&#50;&#55;&#61;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"81\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834583758\">\n<p id=\"fs-id1167834583760\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-9418f76b7fb8efbd61d4b14b3df06bad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#61;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"42\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826864437\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834583596\">\n<p id=\"fs-id1167834583598\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-24fb6573560dac5bf4abe98de97bed0c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#97;&#125;&#51;&#50;&#61;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"80\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167827164772\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167830991254\">\n<p id=\"fs-id1167830991256\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-91d816061b5c9e50ca4e43fdd7f2a96c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#97;&#125;&#50;&#52;&#61;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"81\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826827883\">\n<p id=\"fs-id1167826827885\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-8c8744070c9422f3a5139e7925a6dcf6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#61;&#92;&#115;&#113;&#114;&#116;&#91;&#51;&#93;&#123;&#50;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"67\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832067469\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167832067471\">\n<p id=\"fs-id1167832067473\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-d6d421b588574dc1c76d3dc51639cfdb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#110;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#61;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"60\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167827966785\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167827966787\">\n<p id=\"fs-id1167830961523\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-0f7cd071eefb6b46c3f8736490a13a3a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#110;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#61;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"61\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167830963537\">\n<p id=\"fs-id1167830963540\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-506e1f30b40f2f35aadfc7fc570f75e3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#123;&#101;&#125;&#94;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"49\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826938241\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167826938243\">\n<p id=\"fs-id1167826938246\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-e10b0c3dd99941106c40f630e6e25a9f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#50;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#120;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"129\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831887569\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167831887571\">\n<p id=\"fs-id1167831887573\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-7ff0a4ce4d9718b51615ed5a4836211e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#50;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#120;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"128\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835534261\">\n<p id=\"fs-id1167835534264\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-8ddab230605c435eb8b7408a736d3e77_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"42\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834300196\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834300198\">\n<p id=\"fs-id1167835283172\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-14e5cdeed1a4017067afef7e639de457_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#51;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#120;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"128\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834448631\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834448633\">\n<p id=\"fs-id1167834448635\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-d94b69b45c1e72ba6bd4786b15d7c1c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#51;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#120;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"129\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-33f93315271ffe344065fe74b6d91ec5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#49;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"52\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831882391\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167831882393\">\n<p id=\"fs-id1167831882395\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-0d43d9f5d5577159457e39da00a40b74_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#52;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#120;&#43;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"129\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826967436\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167826967438\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-d9d7b92c5c10436910e4ea0baa2f9ebe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#52;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#120;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"128\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835306013\">\n<p id=\"fs-id1167835306015\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-116dfacb4c2a32bdfb5ab9e852f29986_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"43\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835303330\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835303332\">\n<p id=\"fs-id1167835303334\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-0ea2dfec54c1601969486cb5cd0e897a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#110;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#101;&#125;&#94;&#123;&#52;&#120;&#125;&#61;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"75\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834438873\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835381450\">\n<p id=\"fs-id1167835381452\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-461db0c3f68cd6c41acf6443140ed0fc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#110;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#101;&#125;&#94;&#123;&#50;&#120;&#125;&#61;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"75\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834226205\">\n<p id=\"fs-id1167834226208\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-3573bf1ea4c223bb71878796b2106731_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"43\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834429288\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834429290\">\n<p id=\"fs-id1167834429292\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-660010e9bb49332a14646e7a73def468_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#61;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"72\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834539458\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834539461\">\n<p id=\"fs-id1167834539463\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-151f0cf1ba6bfbf51a8bc39b17b7becc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"131\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835370937\">\n<p id=\"fs-id1167835370939\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-f9280b99f652e996488aa83901a02820_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#53;&#92;&#115;&#113;&#114;&#116;&#123;&#53;&#125;&#44;&#120;&#61;&#53;&#92;&#115;&#113;&#114;&#116;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"154\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167830837484\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167830837486\">\n<p id=\"fs-id1167830837488\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-d499aa48710da152d2e6a548da9c0645_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#50;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"130\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167828411060\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167828411062\">\n<p id=\"fs-id1167828411064\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-736bc0f54cc46ac9cbbd6cb48e820e84_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#51;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"131\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835529056\">\n<p id=\"fs-id1167835529058\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-1d8d70ba07f8c708ffd472fd6033746f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#53;&#44;&#120;&#61;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"106\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167832042112\"><strong data-effect=\"bold\">Use Logarithmic Models in Applications<\/strong><\/p>\n<p id=\"fs-id1167832058625\">In the following exercises, use a logarithmic model to solve.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167832058628\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167832058631\">\n<p id=\"fs-id1167832058633\">What is the decibel level of normal conversation with intensity <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-0974ccf82a8407f27024741f1b87b04a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#49;&#48;&#125;&#94;&#123;&#45;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"35\" style=\"vertical-align: -1px;\" \/> watts per square inch?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826874702\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167826874704\">\n<p id=\"fs-id1167826874706\">What is the decibel level of a whisper with intensity <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-958dc3155f80233aef3179f7872581f4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#49;&#48;&#125;&#94;&#123;&#45;&#49;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"42\" style=\"vertical-align: -1px;\" \/> watts per square inch?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835390194\">\n<p id=\"fs-id1167835390196\">A whisper has a decibel level of 20 dB.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835390201\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835390204\">\n<p id=\"fs-id1167831031076\">What is the decibel level of the noise from a motorcycle with intensity <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-a2adce90e7fd78901b14540423deb0c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#49;&#48;&#125;&#94;&#123;&#45;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"35\" style=\"vertical-align: -1px;\" \/> watts per square inch?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167830866003\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167830866006\">\n<p id=\"fs-id1167831811701\">What is the decibel level of the sound of a garbage disposal with intensity <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-a2adce90e7fd78901b14540423deb0c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#49;&#48;&#125;&#94;&#123;&#45;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"35\" style=\"vertical-align: -1px;\" \/> watts per square inch?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835377631\">\n<p id=\"fs-id1167835377633\">The sound of a garbage disposal has a decibel level of 100 dB.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835377638\" class=\"material-set-2\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167835510386\">In 2014, Chile experienced an intense earthquake with a magnitude of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-17f9334def3aeb2025134bd752d0dea3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#46;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"22\" style=\"vertical-align: 0px;\" \/> on the Richter scale. In 2010, Haiti also experienced an intense earthquake which measured <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-5b3cacec8b1591a62c3825dfdfb15cde_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#46;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"23\" style=\"vertical-align: 0px;\" \/> on the Richter scale. Compare the intensities of the two earthquakes.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834188881\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834188883\">\n<p id=\"fs-id1167834188886\">The Los Angeles area experiences many earthquakes. In 1994, the Northridge earthquake measured magnitude of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-980582f622e2f0d52267dfc2f85f642d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#46;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"23\" style=\"vertical-align: 0px;\" \/> on the Richter scale. In 2014, Los Angeles also experienced an earthquake which measured <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-90ed214187cfe95bbbbe178e3ad0dc3d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#46;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"22\" style=\"vertical-align: -1px;\" \/> on the Richter scale. Compare the intensities of the two earthquakes.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835532713\">\n<p id=\"fs-id1167835532715\">The intensity of the 1994 Northridge earthquake in the Los Angeles area was about 40 times the intensity of the 2014 earthquake.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"writing\" data-depth=\"2\" id=\"fs-id1167831112276\">\n<h4 data-type=\"title\">Writing Exercises<\/h4>\n<div data-type=\"exercise\" id=\"fs-id1167831881048\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167831881050\">\n<p id=\"fs-id1167831881052\">Explain how to change an equation from logarithmic form to exponential form.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835324862\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835324864\">\n<p id=\"fs-id1167835324867\">Explain the difference between common logarithms and natural logarithms.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835324871\">\n<p id=\"fs-id1167835524534\">Answers will vary.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835524540\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835524542\">\n<p id=\"fs-id1167835524544\">Explain why <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-32f9df9c4bd0a56af2ad88dcc493b0e0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#97;&#125;&#123;&#97;&#125;&#94;&#123;&#120;&#125;&#61;&#120;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"86\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831895106\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167831895108\">\n<p id=\"fs-id1167831895110\">Explain how to find the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-50c0c21a6b8e8ad7c1e6e8a64a7ca30e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#55;&#125;&#51;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"47\" style=\"vertical-align: -4px;\" \/> on your calculator.<\/p>\n<\/div>\n<div data-type=\"solution\">\n<p id=\"fs-id1167835216587\">Answers will vary.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167835216593\">\n<h4 data-type=\"title\">Self Check<\/h4>\n<p id=\"fs-id1167834188808\"><span class=\"token\">\u24d0<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p>After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.<span data-type=\"media\" id=\"fs-id1167834188817\" data-alt=\"This table has four rows and five columns. The first row, which serves as a header, reads I can\u2026, Confidently, With some help, and No\u2014I don\u2019t get it. The first column below the header row reads Convert between exponential and logarithmic form, evaluate logarithmic functions, graph logarithmic functions, solve logarithmic equations, and use logarithmic models in applications. The rest of the cells are blank.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcbcintermediatealgebra\/wp-content\/uploads\/sites\/1021\/2020\/05\/CNX_IntAlg_Figure_10_03_201_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This table has four rows and five columns. The first row, which serves as a header, reads I can\u2026, Confidently, With some help, and No\u2014I don\u2019t get it. The first column below the header row reads Convert between exponential and logarithmic form, evaluate logarithmic functions, graph logarithmic functions, solve logarithmic equations, and use logarithmic models in applications. The rest of the cells are blank.\" \/><\/span><\/p>\n<p id=\"fs-id1167835284688\"><span class=\"token\">\u24d1<\/span> After reviewing this checklist, what will you do to become confident for all objectives?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"glossary\" class=\"textbox shaded\">\n<h3 data-type=\"glossary-title\">Glossary<\/h3>\n<dl id=\"fs-id1167831824962\">\n<dt>common logarithmic function<\/dt>\n<dd id=\"fs-id1167831824966\">The function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-7a38f69803eed430423e440fd4880f32_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"97\" style=\"vertical-align: -4px;\" \/> is the common logarithmic function with base<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-86d1572e00759b4a971be10302e346aa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#48;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"21\" style=\"vertical-align: -4px;\" \/> where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-68c3061ebd908643e1b359493c6a0a35_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#62;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"47\" style=\"vertical-align: 0px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"equation\" id=\"fs-id1167826987756\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-3cef176c6374bd4e34e9872b10c2b32f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#105;&#115;&#32;&#101;&#113;&#117;&#105;&#118;&#97;&#108;&#101;&#110;&#116;&#32;&#116;&#111;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#61;&#123;&#49;&#48;&#125;&#94;&#123;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"253\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/dd>\n<\/dl>\n<dl id=\"fs-id1167832125662\">\n<dt>logarithmic function<\/dt>\n<dd id=\"fs-id1167834224743\">The function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-9b7e4e7b3caea150723d3fa578a79d72_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#97;&#125;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"102\" style=\"vertical-align: -4px;\" \/> is the logarithmic function with base <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-d8fa4ffebd0bdc66a5e68aa7b712f46c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"13\" style=\"vertical-align: -4px;\" \/> where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-aaf802bf2187ba09e2c109dce11ba56c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#62;&#48;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"46\" style=\"vertical-align: -4px;\" \/><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-a22c7c2a68ba0f695ec4c27fcf8628d4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#62;&#48;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"47\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-31658a6c5c3f18d97495ba25506408f2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#92;&#110;&#101;&#32;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"46\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"equation\" id=\"fs-id1167830770122\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-7018593e8e8e71bf68f4cf11840968c3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#111;&#103;&#125;&#125;&#95;&#123;&#97;&#125;&#120;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#105;&#115;&#32;&#101;&#113;&#117;&#105;&#118;&#97;&#108;&#101;&#110;&#116;&#32;&#116;&#111;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#61;&#123;&#97;&#125;&#94;&#123;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"249\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/dd>\n<\/dl>\n<dl id=\"fs-id1167834464022\">\n<dt>natural logarithmic function<\/dt>\n<dd id=\"fs-id1167834464025\">The function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-56e1c7d8eea20bd2ebeb56beb8c81bc0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#110;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"89\" style=\"vertical-align: -4px;\" \/> is the natural logarithmic function with base <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-cdd204f9eb9bf059f6c66abbb4af16d5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#101;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"12\" style=\"vertical-align: -4px;\" \/> where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-68c3061ebd908643e1b359493c6a0a35_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#62;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"47\" style=\"vertical-align: 0px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"equation\" id=\"fs-id1167834438819\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-content\/ql-cache\/quicklatex.com-2538f084b4ef090b588044b988a4420b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#110;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#105;&#115;&#32;&#101;&#113;&#117;&#105;&#118;&#97;&#108;&#101;&#110;&#116;&#32;&#116;&#111;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#61;&#123;&#101;&#125;&#94;&#123;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"235\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/dd>\n<\/dl>\n<\/div>\n","protected":false},"author":9,"menu_order":148,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-4158","chapter","type-chapter","status-publish","hentry"],"part":3993,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-json\/pressbooks\/v2\/chapters\/4158","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-json\/wp\/v2\/users\/9"}],"version-history":[{"count":1,"href":"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-json\/pressbooks\/v2\/chapters\/4158\/revisions"}],"predecessor-version":[{"id":4159,"href":"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-json\/pressbooks\/v2\/chapters\/4158\/revisions\/4159"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-json\/pressbooks\/v2\/parts\/3993"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-json\/pressbooks\/v2\/chapters\/4158\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-json\/wp\/v2\/media?parent=4158"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-json\/pressbooks\/v2\/chapter-type?post=4158"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-json\/wp\/v2\/contributor?post=4158"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/jkdcintermediatealgebracloned\/wp-json\/wp\/v2\/license?post=4158"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}