{"id":407,"date":"2019-04-29T12:35:30","date_gmt":"2019-04-29T16:35:30","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/?post_type=back-matter&p=407"},"modified":"2019-04-29T12:36:01","modified_gmt":"2019-04-29T16:36:01","slug":"reference-section","status":"publish","type":"back-matter","link":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/back-matter\/reference-section\/","title":{"raw":"Reference Section","rendered":"Reference Section"},"content":{"raw":"Symbols & Abbreviations\r\nCommon Powers\r\nSI Unit Prefixes\r\nGreek Alphabet\r\nLinear Inequalities\r\nProperties of Absolute Values\r\nMetric to English (US) Conversions\r\nPlane Geometry Formula\r\nSolid Geometry Formula\r\nPythagorean Theorem (Variations)\r\nLinear Equations\r\nConic Sections\r\nPolynomials, Pascals Triangle\r\nProperties of Complex Numbers\r\nExponents & Radicals\r\nTrigonometric Functions & Values\r\nTrigonometric Identities\r\nBasic Trigonometric Ratios Graphs\r\nTrigonometric Tables\r\nProperties of Logarithmic Functions\r\nCommon Logarithmic Tables\r\n\r\nCommon Powers\r\n(and not so common)\r\n\r\nSquares Cubes 4th Power 5th Power 6th Power 7th Power\r\n22 = 4 23 = 8 24 = 16 25 = 32 26 = 64 27 = 128\r\n32 = 9 33 = 27 34 = 81 35 = 243 36 = 729 37 = 2187\r\n42 = 16 43 = 64 44 = 256 45 = 1024 46 = 4096 47 = 16384\r\n52 = 25 53 = 125 54 = 625 55 = 3125 56 = 15625 57 = 78125\r\n62 = 36 63 = 216 64 = 1296 65 = 7776 66 = 46656 67 = 279936\r\n72 = 49 73 = 343 74 = 2401 75 = 16807 76 = 117649 77 = 823543\r\n82 = 64 83 = 512 84 = 4096 85 = 32768 86 = 262144 87 = 2097152\r\n92 = 81 93 = 729 94 = 6561 95 = 59049 96 = 531441 97 = 4782969\r\n102 = 100 103 = 1000 104 = 10000 105 = 100000 106 = 1000000 107 = 10000000\r\n\r\n112 = 121 122 = 144 132 = 169 142 = 196 152 = 225 202 = 400\r\n\r\nGreek Alphabet\r\n\r\nA \u03b1 Alpha N v Nu\r\nB \u03b2 Beta \u039e \u03be Xi\r\n\u0393 \u03b3 Gamma O o Omicron\r\n\u2206 \u03b4 Delta \u220f \u03c0 Pi\r\nE \u03b5 Epsilon P \u03c1 Rho\r\nZ \u03b6 Zeta \u2211 \u03c3 Sigma\r\nH \u03b7 Eta T \u03c4 Tau\r\n\u0398 \u03b8 Theta \u03a5 v Upsilon\r\nI i Iota \u03a6 \u03c6 Phi\r\nK k Kappa \u03a7 \u03c7 Chi\r\n\u039b \u03bb Lambda \u03a8 \u03c8 Psi\r\nM \u00b5 Mu \u03a9 \u03c9 Omega\r\n\r\nSI Unit Prefixes\r\n\r\nFactor Name Symbol Factor Name Symbol\r\n10-18 atto a 10-1 deci d\r\n10-15 femto f 10 deca da\r\n10-12 pico p 102 hecto h\r\n10-9 nano n 103 kilo k\r\n10-6 micro \u00b5 106 mega M\r\n10-3 milli m 109 giga G\r\n10-2 centi c 1012 tera T\r\nLinear Inequalities\r\nInterval Notation Set Builder Notation Graph of the Inequality\r\n\r\n(a, + \u221e) {x | x > a}\r\n\r\n[a, + \u221e) {x | x \u2265 a}\r\n\r\n(- \u221e, a) {x | x < a}\r\n\r\n(- \u221e, a] {x | x \u2264 a}\r\n\r\n[a, b] {x | a \u2264 x \u2264 b}\r\n\r\n(a, b) {x | a < x < b}\r\n\r\n[a, b) {x | a \u2264 x < b}\r\n\r\n(a, b] {x | a < x \u2264 b}\r\n\r\n(- \u221e, + \u221e) {x | x \u2208 R}\r\n(- \u221e, b) or (a, + \u221e) {x | x < a or x > b}\r\n\r\n(- \u221e, a] or [a, + \u221e) {x | x < a or x > b}\r\n\r\n(- \u221e, a] or [a, + \u221e) {x | x < a or x > b}\r\n\r\n(- \u221e, a] or [a, + \u221e) {x | x < a or x > b}\r\n\r\nProperties of Absolute Values\r\nIf | X | = k, then X = k or X = -k\r\nIf | X | < k, then -k < X < k\r\nIf | X | > k, then X > k or X < -kMetric to English (US) Conversions\r\n\r\nDistance:\r\n12 in = 1 ft\r\n3 ft = 1 yd 10 mm = 1 cm\r\n1760 yds = 1 mi 100 cm = 1 m\r\n5280 ft = 1 mi 1000m = 1 km\r\n(English-Metric conversions: 1 inch = 2.54 cm; 1 mile = 1.61 km)\r\n\r\nArea:\r\n144 in2 = 1 ft2 10,000 cm2 = 1 m2\r\n43,560 ft2 = 1 acre 10,000 m2 = 1 hectare\r\n640 acres = 1 mi2 100 hectare = 1 km2\r\n(English-Metric conversions: 1 in2 = 6.45 cm2; 1 mi2 = 2.59 km2)\r\n\r\nVolume:\r\n57.75 in3 = 1 qt 1 cm3 = 1 ml\r\n4 qt = 1 gal 1000 ml = 1 liter\r\n42 gal (petroleum) = 1 barrel 1000 liter = 1 m3\r\n(English-Metric conversions: 16.39 cm3 = 1 in3; 3.79 liters = 1 gal)\r\n\r\nMass:\r\n437.5 grains = 1 oz 1000 mg = 1 g\r\n16 oz = 1 lb 1000 g = 1 kg\r\n2000 lb = 1 short ton 1000 kg = 1 metric ton\r\n(English-Metric conversions: 453 g = 1 lb; 2.2 lb = 1 kg)\r\n\r\nTemperature:\r\n(Fahrenheit - Celsius Conversions: \u00b0C = 5\/9 (\u00b0F - 32) and \u00b0F = 9\/5 \u00b0C + 32)\r\n\r\nPlane Geometry Formula\r\n\r\nCircle Square Rectangle\r\nArea = \u03c0 r2 Area = s2 Area = l w\r\nPerimeter = 2 \u03c0 r Perimeter = 4 s Perimeter = 2l + 2w\r\n\r\nTriangle Rhombus Trapezoid\r\nArea = 1\/2 b h Area = b h Area = 1\/2 (l1 + l2) h\r\nPerimeter = s1 + s2 + s3 Perimeter = 4 b Perimeter = l1 + l2 + h1 + h2\r\nParallelogram Regular Polygon (n-gon)\r\nArea = b h Area = (1\/2 s h) (number of sides)\r\nPerimeter = 2 h1 + 2 b Perimeter = s (number of sides)\r\nSolid Geometry Formula\r\nCube Right Rectangular Prism Right Cylindrical Prism\r\nVolume = s3 Volume = l w h Volume = \u03c0 r2 h\r\nS. A. = 6 s2 S. A. = 2 l w + 2 h w + 2 l h S. A. = 2 \u03c0 r h + 2 \u03c0 r2\r\n\r\nSphere Torus Right Triangular Prism\r\nVolume = 4\/3 \u03c0 r3 Volume = 2 \u03c02 r2 R Volume = (1\/2 b h) l\r\nS. A. = 4 \u03c0 r2 S. A. = 4 \u03c02 r R S. A. = b h + 2 l s + l b\r\nRight Circular Cone General Cone\/Pyramid Square Pyramid\r\nVolume = 1\/3 (\u03c0 r2) h Volume = 1\/3 (base area) h Volume = 1\/3 (s2) h\r\nS. A. = \u03c0 r (r2 + h2)1\/2 + \u03c0 r2 S.A. = s [s + (s2 + 4h2)\r\nPythagorean Theorem (Variations)\r\n\r\nFor any right triangle a, b and c:\r\n\r\na2 + b2 = c2\r\n\r\nFor any non-right triangle a, b and c:\r\n\r\na2 = b2 + c2 - 2bc cos A\r\nb2 = a2 + c2 - 2ac cos B\r\nc2 = a2 + b2 - 2ab cos C\r\n\r\nFor any rectangular prism a, b and c, the diagonal (d) length is:\r\n\r\nd2 = a2 + b2 + c2\r\n\r\nLinear Equations\r\nAn Ordered Pair: (x, y)\r\nDistance between Two Ordered Pairs: d2 = \u2206 x2 + \u2206 y2 or d2 = (x2 - x1)2 + (y2 - y1)2\r\nMidpoint between Two Ordered Pairs: [(x1 + x2) , (y1 + y2)]\r\n2 2\r\nSlope: m = \u2206 y or m = (y2 - y1) ... where \u2206y = y2 - y1, \u2206 x = x2 - x1 and \u2206 x \u2260 0\r\n\u2206 x (x2 - x1)\r\nThe slope for Two Parallel Lines: m1 = m2\r\nThe slope for Two Perpendicular Lines: m1 . m2 = -1 or m1 = -1\/m2\r\nTo find the Linear Equation Using Two Ordered Pairs: (x2 - x1) m = (y2 - y1)\r\nGeneral Form of a Linear Equation: Ax + By + C = 0 (A, B, C are integers, A is positive)\r\nSlope Intercept Form of a Linear Equation: y = mx + b\r\nConic Sections\r\n\r\nConic Equations (Standard Form):\r\n\r\nCircle: (x - h)2 + (y - k)2 = r2 (h, k) is the center point, r is the radius from the\r\ncenter to the circles (x, y) coordinates\r\n\r\nParabolas: y - k = a(x - h)2 Parabolas, commonly written as y = ax2 + bx + c\r\nx - h = a(y - k)2\r\n\r\nEllipse: (x - h)2 + (y - k)2 = 1 (h, k) is the center point, rx is the radius length in rx2 ry2 the \u00b1 x direction, ry is the radius length in the \u00b1 y direction\r\n\r\nHyperbola: (x - h)2 - (y - k)2 = 1 (h, k) is the center point, rx is the distance from the\r\nrx2 ry2 center to the hyperbola\u2019s \u00b1 x asymptote. ry is the distance from the center to the hyperbola\u2019s \u00b1 x asymptote.Polynomials\r\n\r\nQuadratic Solutions:\r\n\r\nThe solution for x from a quadratic equation ax2 + bx + c = 0, (where a \u2260 0), can be found from:\r\n\r\nFactoring:\r\na2 - b2 = (a + b)(a - b) a2 + b2 ... cannot be factored\r\na3 - b3 = (a - b)(a2 + ab + b2) a3 + b3 = (a + b)(a2 - ab + b2)\r\n\r\nBinomial Expansions:\r\n(a + b)0 = 1 (a - b)0 = 1\r\n(a + b)1 = a + b (a - b)1 = a - b\r\n(a + b)2 = a2 + 2ab + b2 (a - b)2 = a2 - 2ab + b2\r\n(a + b)3 = a3 + 3a2b + 3ab2 + b3 (a - b)3 = a3 - 3a2b + 3ab2 - b3\r\n(a + b)4 = a4 + 4a3b + 6a2b2 + 4ab3 + b4 (a - b)4 = a4 - 4a3b + 6a2b2 - 4ab3 + b4\r\n(a + b)5 = a5 + 5a4b + 10a3b2 + 10a2b3 + 5ab4 + b5\r\n(a - b)5 = a5 - 5a4b + 10a3b2 - 10a2b3 + 5ab4 - b5\r\n(a + b)6 = a6 + 6a5b + 15a4b2 + 20a3b3 + 15a2b4 + 6ab5 + b5\r\n(a - b)6 = a6 - 6a5b + 15a4b2 - 20a3b3 + 15a2b4 - 6ab5 + b5\r\n\r\nProperties of Complex Numbers\r\n\r\n( a + bi) + (c + di) = a + c + (b + d)i ( a + bi) - (c + di) = a - c + (b - d)i\r\n\r\n( a + bi) (c + di) = ac - bd + (ab + bd)i ( a + bi) (a - bi) = a2 + b2\r\n\r\n(-a)1\/2 = i(a)1\/2, a \u2265 0\r\n\r\nProperties of Exponents\r\n\r\nProperties of Rational Exponents and Radicals\r\n\r\nBasic Trigonometric Functions & Values\r\n\r\nBasic Trigonometric Ratios\r\n\r\nSin = Opposite Cos = Adjacent Tan = Opposite\r\nHypotenuse Hypotenuse Adjacent\r\n\r\nSec = Hypotenuse Csc = Hypotenuse Cot = Adjacent\r\nOpposite Adjacent Opposite\r\n\r\nTrigonometric Identities\r\n\r\nReciprocal Identities:\r\nsin \u03b8 = 1\/csc \u03b8 tan \u03b8 = 1\/cot \u03b8 cos \u03b8 = 1\/sec \u03b8\r\ncsc \u03b8 = 1\/sin \u03b8 cot \u03b8 = 1\/tan \u03b8 sec \u03b8 = 1\/cos \u03b8\r\n\r\nTangent and Cotangent Identities:\r\ntan \u03b8 = sin \u03b8 \/ cos \u03b8 cot \u03b8 = cos \u03b8 \/ sin \u03b8\r\n\r\nPythagorean Identities:\r\nsin2 \u03b8 + cos2 \u03b8 =1 tan2 \u03b8 + 1 = sec2 \u03b8 1 + cot2 \u03b8 = csc2 \u03b8\r\n\r\nDouble Angle Formulas:\r\nsin 2\u03b8 = 2 sin \u03b8 cos \u03b8\r\ncos 2\u03b8 = cos2 \u03b8 \u2212 sin2 \u03b8 cos 2\u03b8 = 2 cos2 \u03b8 \u2212 1 cos 2\u03b8 = 1\u2212 2sin2 \u03b8\r\ntan 2\u03b8 = 2 tan \u03b8 \/ 1\u2212 tan2 \u03b8\r\n\r\nSum and Difference Formulas:\r\nsin (\u03b1 + \u03b2 ) = sin \u03b1 cos \u03b2 + cos \u03b1 sin \u03b2 sin (\u03b1 \u2212 \u03b2 ) = sin \u03b1 cos \u03b2 \u2212 cos \u03b1 sin \u03b2\r\ncos (\u03b1 + \u03b2 ) = cos \u03b1 cos \u03b2 \u2212 sin \u03b1 sin \u03b2 cos (\u03b1 \u2212 \u03b2 ) = cos \u03b1 cos \u03b2 + sin \u03b1 sin \u03b2\r\ntan(\u03b1 + \u03b2) = tan \u03b1 + tan \u03b2 \/ 1 \u2212 tan \u03b1 tan \u03b2 tan(\u03b1 \u2212 \u03b2) = tan \u03b1 \u2212 tan \u03b2 \/ 1 + tan \u03b1 tan \u03b2\r\n\r\nGraphs of Basic Trigonometric Ratios\r\nTrigonometric Tables\r\n\r\nAngle Sin Cos Tan Csc Angle Sin Cos Tan Csc\r\n1 0.017 1.000 0.017 57.299 46 0.719 0.695 1.036 1.390\r\n2 0.035 0.999 0.035 28.654 47 0.731 0.682 1.072 1.36\r\n3 0.052 0.999 0.052 19.107 48 0.743 0.669 1.111 1.346\r\n4 0.070 0.998 0.070 14.336 49 0.755 0.656 1.150 1.325\r\n5 0.087 0.996 0.087 11.474 50 0.766 0.643 1.192 1.305\r\n6 0.105 0.995 0.105 9.567 51 0.777 0.629 1.235 1.287\r\n7 0.122 0.993 0.123 8.206 52 0.788 0.616 1.280 1.269\r\n8 0.139 0.990 0.141 7.185 53 0.799 0.602 1.327 1.252\r\n9 0.156 0.988 0.158 6.392 54 0.809 0.588 1.376 1.236\r\n10 0.174 0.985 0.176 5.759 55 0.819 0.574 1.428 1.221\r\n11 0.191 0.982 0.194 5.241 56 0.829 0.559 1.483 1.206\r\n12 0.208 0.978 0.213 4.810 57 0.839 0.545 1.540 1.192\r\n13 0.225 0.974 0.231 4.445 58 0.848 0.530 1.600 1.179\r\n14 0.242 0.970 0.249 4.134 59 0.857 0.515 1.664 1.167\r\n15 0.259 0.966 0.268 3.864 60 0.866 0.500 1.732 1.155\r\n16 0.276 0.961 0.287 3.628 61 0.875 0.485 1.804 1.143\r\n17 0.292 0.956 0.306 3.420 62 0.883 0.469 1.881 1.133\r\n18 0.309 0.951 0.325 3.236 63 0.891 0.454 1.963 1.122\r\n19 0.326 0.946 0.344 3.072 64 0.899 0.438 2.050 1.113\r\n20 0.342 0.940 0.364 2.924 65 0.906 0.423 2.145 1.103\r\n21 0.358 0.934 0.384 2.790 66 0.914 0.407 2.246 1.095\r\n22 0.375 0.927 0.404 2.669 67 0.921 0.391 2.356 1.086\r\n23 0.391 0.921 0.424 2.559 68 0.927 0.375 2.475 1.079\r\n24 0.407 0.914 0.445 2.459 69 0.934 0.358 2.605 1.071\r\n25 0.423 0.906 0.466 2.366 70 0.940 0.342 2.747 1.064\r\n26 0.438 0.899 0.488 2.281 71 0.946 0.326 2.904 1.058\r\n27 0.454 0.891 0.510 2.203 72 0.951 0.309 3.078 1.051\r\n28 0.469 0.883 0.532 2.130 73 0.956 0.292 3.271 1.046\r\n29 0.485 0.875 0.554 2.063 74 0.961 0.276 3.487 1.040\r\n30 0.500 0.866 0.577 2.000 75 0.966 0.259 3.732 1.035\r\n31 0.515 0.857 0.601 1.942 76 0.970 0.242 4.011 1.031\r\n32 0.530 0.848 0.625 1.887 77 0.974 0.225 4.331 1.026\r\n33 0.545 0.839 0.649 1.836 78 0.978 0.208 4.705 1.022\r\n34 0.559 0.829 0.675 1.788 79 0.982 0.191 5.145 1.019\r\n35 0.574 0.819 0.700 1.743 80 0.985 0.174 5.671 1.015\r\n36 0.588 0.809 0.727 1.701 81 0.988 0.156 6.314 1.012\r\n37 0.602 0.799 0.754 1.662 82 0.990 0.139 7.115 1.010\r\n38 0.616 0.788 0.781 1.624 83 0.993 0.122 8.144 1.008\r\n39 0.629 0.777 0.810 1.589 84 0.995 0.105 9.514 1.006\r\n40 0.643 0.766 0.839 1.556 85 0.996 0.087 11.430 1.004\r\n41 0.656 0.755 0.869 1.524 86 0.998 0.070 14.301 1.002\r\n42 0.669 0.743 0.900 1.494 87 0.999 0.052 19.081 1.001\r\n43 0.682 0.731 0.933 1.466 88 0.999 0.035 28.636 1.001\r\n44 0.695 0.719 0.966 1.440 89 1.000 0.017 57.290 1.000\r\n45 0.707 0.707 1.000 1.414 90 1.000 0.000 \u00a0 1.000 Properties of Logarithmic Functions\r\n\r\nx = ay is equivalent to y = loga x ex = y is equivalent to ln y = x\r\nloga (xy) = loga x + loga y loga (x\/y) = loga x - loga y loga (1\/x) = - loga x\r\n\r\nln (xy) = ln x - ln y ln (x\/y) = ln x - ln y ln (1\/x) = - ln x\r\n\r\nloga x = log x\/ log a loga a = 1 loga 1 = 0 loga xy = y loga x\r\n\r\nloga x = ln x\/ ln a ln e = 1 ln 1 = 0 ln xy = y ln x\r\n\r\nCommon Logarithm Table\r\nN 0 1 2 3 4 5 6 7 8 9\r\n1.0 0.0000 0.0043 0.0086 0.0128 0.0170 0.0212 0.0253 0.0294 0.0334 0.0374\r\n1.1 0.0414 0.0453 0.0492 0.0531 0.0569 0.0607 0.0645 0.0682 0.0719 0.0755\r\n1.2 0.0792 0.0828 0.0864 0.0899 0.0934 0.0969 0.1004 0.1038 0.1072 0.1106\r\n1.3 0.1139 0.1173 0.1206 0.1239 0.1271 0.1303 0.1335 0.1367 0.1399 0.1430\r\n1.4 0.1461 0.1492 0.1523 0.1553 0.1584 0.1614 0.1644 0.1673 0.1703 0.1732\r\n1.5 0.1761 0.1790 0.1818 0.1847 0.1875 0.1903 0.1931 0.1959 0.1987 0.2014\r\n1.6 0.2041 0.2068 0.2095 0.2122 0.2148 0.2175 0.2201 0.2227 0.2253 0.2279\r\n1.7 0.2304 0.2330 0.2355 0.2380 0.2405 0.2430 0.2455 0.2480 0.2504 0.2529\r\n1.8 0.2553 0.2577 0.2601 0.2625 0.2648 0.2672 0.2695 0.2718 0.2742 0.2765\r\n1.9 0.2788 0.2810 0.2833 0.2856 0.2878 0.2900 0.2923 0.2945 0.2967 0.2989\r\n2.0 0.3010 0.3032 0.3054 0.3075 0.3096 0.3118 0.3139 0.3160 0.3181 0.3201\r\n2.1 0.3222 0.3243 0.3263 0.3284 0.3304 0.3324 0.3345 0.3365 0.3385 0.3404\r\n2.2 0.3424 0.3444 0.3464 0.3483 0.3502 0.3522 0.3541 0.3560 0.3579 0.3598\r\n2.3 0.3617 0.3636 0.3655 0.3674 0.3692 0.3711 0.3729 0.3747 0.3766 0.3784\r\n2.4 0.3802 0.3820 0.3838 0.3856 0.3874 0.3892 0.3909 0.3927 0.3945 0.3962\r\n2.5 0.3979 0.3997 0.4014 0.4031 0.4048 0.4065 0.4082 0.4099 0.4116 0.4133\r\n2.6 0.4150 0.4166 0.4183 0.4200 0.4216 0.4232 0.4249 0.4265 0.4281 0.4298\r\n2.7 0.4314 0.4330 0.4346 0.4362 0.4378 0.4393 0.4409 0.4425 0.4440 0.4456\r\n2.8 0.4472 0.4487 0.4502 0.4518 0.4533 0.4548 0.4564 0.4579 0.4594 0.4609\r\n2.9 0.4624 0.4639 0.4654 0.4669 0.4683 0.4698 0.4713 0.4728 0.4742 0.4757\r\n3.0 0.4771 0.4786 0.4800 0.4814 0.4829 0.4843 0.4857 0.4871 0.4886 0.4900\r\nN 0 1 2 3 4 5 6 7 8 9\r\n3.1 0.4914 0.4928 0.4942 0.4955 0.4969 0.4983 0.4997 0.5011 0.5024 0.5038\r\n3.2 0.5051 0.5065 0.5079 0.5092 0.5105 0.5119 0.5132 0.5145 0.5159 0.5172\r\n3.3 0.5185 0.5198 0.5211 0.5224 0.5237 0.5250 0.5263 0.5276 0.5289 0.5302\r\n3.4 0.5315 0.5328 0.5340 0.5353 0.5366 0.5378 0.5391 0.5403 0.5416 0.5428\r\n3.5 0.5441 0.5453 0.5465 0.5478 0.5490 0.5502 0.5514 0.5527 0.5539 0.5551\r\n3.6 0.5563 0.5575 0.5587 0.5599 0.5611 0.5623 0.5635 0.5647 0.5658 0.5670\r\n3.7 0.5682 0.5694 0.5705 0.5717 0.5729 0.5740 0.5752 0.5763 0.5775 0.5786\r\n3.8 0.5798 0.5809 0.5821 0.5832 0.5843 0.5855 0.5866 0.5877 0.5888 0.5899\r\n3.9 0.5911 0.5922 0.5933 0.5944 0.5955 0.5966 0.5977 0.5988 0.5999 0.6010\r\n4.0 0.6021 0.6031 0.6042 0.6053 0.6064 0.6075 0.6085 0.6096 0.6107 0.6117\r\n4.1 0.6128 0.6138 0.6149 0.6160 0.6170 0.6180 0.6191 0.6201 0.6212 0.6222\r\n4.2 0.6232 0.6243 0.6253 0.6263 0.6274 0.6284 0.6294 0.6304 0.6314 0.6325\r\n4.3 0.6335 0.6345 0.6355 0.6365 0.6375 0.6385 0.6395 0.6405 0.6415 0.6425\r\n4.4 0.6435 0.6444 0.6454 0.6464 0.6474 0.6484 0.6493 0.6503 0.6513 0.6522\r\n4.5 0.6532 0.6542 0.6551 0.6561 0.6571 0.6580 0.6590 0.6599 0.6609 0.6618\r\n4.6 0.6628 0.6637 0.6646 0.6656 0.6665 0.6675 0.6684 0.6693 0.6702 0.6712\r\n4.7 0.6721 0.6730 0.6739 0.6749 0.6758 0.6767 0.6776 0.6785 0.6794 0.6803\r\n4.8 0.6812 0.6821 0.6830 0.6839 0.6848 0.6857 0.6866 0.6875 0.6884 0.6893\r\n4.9 0.6902 0.6911 0.6920 0.6928 0.6937 0.6946 0.6955 0.6964 0.6972 0.6981\r\n5.0 0.6990 0.6998 0.7007 0.7016 0.7024 0.7033 0.7042 0.7050 0.7059 0.7067\r\n5.1 0.7076 0.7084 0.7093 0.7101 0.7110 0.7118 0.7126 0.7135 0.7143 0.7152\r\n5.2 0.7160 0.7168 0.7177 0.7185 0.7193 0.7202 0.7210 0.7218 0.7226 0.7235\r\n5.3 0.7243 0.7251 0.7259 0.7267 0.7275 0.7284 0.7292 0.7300 0.7308 0.7316\r\n\r\nN 0 1 2 3 4 5 6 7 8 9\r\n5.4 0.7324 0.7332 0.7340 0.7348 0.7356 0.7364 0.7372 0.7380 0.7388 0.7396\r\n5.5 0.7404 0.7412 0.7419 0.7427 0.7435 0.7443 0.7451 0.7459 0.7466 0.7474\r\n5.6 0.7482 0.7490 0.7497 0.7505 0.7513 0.7520 0.7528 0.7536 0.7543 0.7551\r\n5.7 0.7559 0.7566 0.7574 0.7582 0.7589 0.7597 0.7604 0.7612 0.7619 0.7627\r\n5.8 0.7634 0.7642 0.7649 0.7657 0.7664 0.7672 0.7679 0.7686 0.7694 0.7701\r\n5.9 0.7709 0.7716 0.7723 0.7731 0.7738 0.7745 0.7752 0.7760 0.7767 0.7774\r\n6.0 0.7782 0.7789 0.7796 0.7803 0.7810 0.7818 0.7825 0.7832 0.7839 0.7846\r\n6.1 0.7853 0.7860 0.7868 0.7875 0.7882 0.7889 0.7896 0.7903 0.7910 0.7917\r\n6.2 0.7924 0.7931 0.7938 0.7945 0.7952 0.7959 0.7966 0.7973 0.7980 0.7987\r\n6.3 0.7993 0.8000 0.8007 0.8014 0.8021 0.8028 0.8035 0.8041 0.8048 0.8055\r\n6.4 0.8062 0.8069 0.8075 0.8082 0.8089 0.8096 0.8102 0.8109 0.8116 0.8122\r\n6.5 0.8129 0.8136 0.8142 0.8149 0.8156 0.8162 0.8169 0.8176 0.8182 0.8189\r\n6.6 0.8195 0.8202 0.8209 0.8215 0.8222 0.8228 0.8235 0.8241 0.8248 0.8254\r\n6.7 0.8261 0.8267 0.8274 0.8280 0.8287 0.8293 0.8299 0.8306 0.8312 0.8319\r\n6.8 0.8325 0.8331 0.8338 0.8344 0.8351 0.8357 0.8363 0.8370 0.8376 0.8382\r\n6.9 0.8388 0.8395 0.8401 0.8407 0.8414 0.8420 0.8426 0.8432 0.8439 0.8445\r\n7.0 0.8451 0.8457 0.8463 0.8470 0.8476 0.8482 0.8488 0.8494 0.8500 0.8506\r\n7.1 0.8513 0.8519 0.8525 0.8531 0.8537 0.8543 0.8549 0.8555 0.8561 0.8567\r\n7.2 0.8573 0.8579 0.8585 0.8591 0.8597 0.8603 0.8609 0.8615 0.8621 0.8627\r\n7.3 0.8633 0.8639 0.8645 0.8651 0.8657 0.8663 0.8669 0.8675 0.8681 0.8686\r\n7.4 0.8692 0.8698 0.8704 0.8710 0.8716 0.8722 0.8727 0.8733 0.8739 0.8745\r\n7.5 0.8751 0.8756 0.8762 0.8768 0.8774 0.8779 0.8785 0.8791 0.8797 0.8802\r\n7.6 0.8808 0.8814 0.8820 0.8825 0.8831 0.8837 0.8842 0.8848 0.8854 0.8859\r\nN 0 1 2 3 4 5 6 7 8 9\r\n7.7 0.8865 0.8871 0.8876 0.8882 0.8887 0.8893 0.8899 0.8904 0.8910 0.8915\r\n7.8 0.8921 0.8927 0.8932 0.8938 0.8943 0.8949 0.8954 0.8960 0.8965 0.8971\r\n7.9 0.8976 0.8982 0.8987 0.8993 0.8998 0.9004 0.9009 0.9015 0.9020 0.9025\r\n8.0 0.9031 0.9036 0.9042 0.9047 0.9053 0.9058 0.9063 0.9069 0.9074 0.9079\r\n8.1 0.9085 0.9090 0.9096 0.9101 0.9106 0.9112 0.9117 0.9122 0.9128 0.9133\r\n8.2 0.9138 0.9143 0.9149 0.9154 0.9159 0.9165 0.9170 0.9175 0.9180 0.9186\r\n8.3 0.9191 0.9196 0.9201 0.9206 0.9212 0.9217 0.9222 0.9227 0.9232 0.9238\r\n8.4 0.9243 0.9248 0.9253 0.9258 0.9263 0.9269 0.9274 0.9279 0.9284 0.9289\r\n8.5 0.9294 0.9299 0.9304 0.9309 0.9315 0.9320 0.9325 0.9330 0.9335 0.9340\r\n8.6 0.9345 0.9350 0.9355 0.9360 0.9365 0.9370 0.9375 0.9380 0.9385 0.9390\r\n8.7 0.9395 0.9400 0.9405 0.9410 0.9415 0.9420 0.9425 0.9430 0.9435 0.9440\r\n8.8 0.9445 0.9450 0.9455 0.9460 0.9465 0.9469 0.9474 0.9479 0.9484 0.9489\r\n8.9 0.9494 0.9499 0.9504 0.9509 0.9513 0.9518 0.9523 0.9528 0.9533 0.9538\r\n9.0 0.9542 0.9547 0.9552 0.9557 0.9562 0.9566 0.9571 0.9576 0.9581 0.9586\r\n9.1 0.9590 0.9595 0.9600 0.9605 0.9609 0.9614 0.9619 0.9624 0.9628 0.9633\r\n9.2 0.9638 0.9643 0.9647 0.9652 0.9657 0.9661 0.9666 0.9671 0.9675 0.9680\r\n9.3 0.9685 0.9689 0.9694 0.9699 0.9703 0.9708 0.9713 0.9717 0.9722 0.9727\r\n9.4 0.9731 0.9736 0.9741 0.9745 0.9750 0.9754 0.9759 0.9763 0.9768 0.9773\r\n9.5 0.9777 0.9782 0.9786 0.9791 0.9795 0.9800 0.9805 0.9809 0.9814 0.9818\r\n9.6 0.9823 0.9827 0.9832 0.9836 0.9841 0.9845 0.9850 0.9854 0.9859 0.9863\r\n9.7 0.9868 0.9872 0.9877 0.9881 0.9886 0.9890 0.9894 0.9899 0.9903 0.9908\r\n9.8 0.9912 0.9917 0.9921 0.9926 0.9930 0.9934 0.9939 0.9943 0.9948 0.9952\r\n9.9 0.9956 0.9961 0.9965 0.9969 0.9974 0.9978 0.9983 0.9987 0.9991 0.9996","rendered":"

Symbols & Abbreviations
\nCommon Powers
\nSI Unit Prefixes
\nGreek Alphabet
\nLinear Inequalities
\nProperties of Absolute Values
\nMetric to English (US) Conversions
\nPlane Geometry Formula
\nSolid Geometry Formula
\nPythagorean Theorem (Variations)
\nLinear Equations
\nConic Sections
\nPolynomials, Pascals Triangle
\nProperties of Complex Numbers
\nExponents & Radicals
\nTrigonometric Functions & Values
\nTrigonometric Identities
\nBasic Trigonometric Ratios Graphs
\nTrigonometric Tables
\nProperties of Logarithmic Functions
\nCommon Logarithmic Tables<\/p>\n

Common Powers
\n(and not so common)<\/p>\n

Squares Cubes 4th Power 5th Power 6th Power 7th Power
\n22 = 4 23 = 8 24 = 16 25 = 32 26 = 64 27 = 128
\n32 = 9 33 = 27 34 = 81 35 = 243 36 = 729 37 = 2187
\n42 = 16 43 = 64 44 = 256 45 = 1024 46 = 4096 47 = 16384
\n52 = 25 53 = 125 54 = 625 55 = 3125 56 = 15625 57 = 78125
\n62 = 36 63 = 216 64 = 1296 65 = 7776 66 = 46656 67 = 279936
\n72 = 49 73 = 343 74 = 2401 75 = 16807 76 = 117649 77 = 823543
\n82 = 64 83 = 512 84 = 4096 85 = 32768 86 = 262144 87 = 2097152
\n92 = 81 93 = 729 94 = 6561 95 = 59049 96 = 531441 97 = 4782969
\n102 = 100 103 = 1000 104 = 10000 105 = 100000 106 = 1000000 107 = 10000000<\/p>\n

112 = 121 122 = 144 132 = 169 142 = 196 152 = 225 202 = 400<\/p>\n

Greek Alphabet<\/p>\n

A \u03b1 Alpha N v Nu
\nB \u03b2 Beta \u039e \u03be Xi
\n\u0393 \u03b3 Gamma O o Omicron
\n\u2206 \u03b4 Delta \u220f \u03c0 Pi
\nE \u03b5 Epsilon P \u03c1 Rho
\nZ \u03b6 Zeta \u2211 \u03c3 Sigma
\nH \u03b7 Eta T \u03c4 Tau
\n\u0398 \u03b8 Theta \u03a5 v Upsilon
\nI i Iota \u03a6 \u03c6 Phi
\nK k Kappa \u03a7 \u03c7 Chi
\n\u039b \u03bb Lambda \u03a8 \u03c8 Psi
\nM \u00b5 Mu \u03a9 \u03c9 Omega<\/p>\n

SI Unit Prefixes<\/p>\n

Factor Name Symbol Factor Name Symbol
\n10-18 atto a 10-1 deci d
\n10-15 femto f 10 deca da
\n10-12 pico p 102 hecto h
\n10-9 nano n 103 kilo k
\n10-6 micro \u00b5 106 mega M
\n10-3 milli m 109 giga G
\n10-2 centi c 1012 tera T
\nLinear Inequalities
\nInterval Notation Set Builder Notation Graph of the Inequality<\/p>\n

(a, + \u221e) {x | x > a}<\/p>\n

[a, + \u221e) {x | x \u2265 a}<\/p>\n

(- \u221e, a) {x | x < a}<\/p>\n

(- \u221e, a] {x | x \u2264 a}<\/p>\n

[a, b] {x | a \u2264 x \u2264 b}<\/p>\n

(a, b) {x | a < x < b}<\/p>\n

[a, b) {x | a \u2264 x < b}<\/p>\n

(a, b] {x | a < x \u2264 b}<\/p>\n

(- \u221e, + \u221e) {x | x \u2208 R}
\n(- \u221e, b) or (a, + \u221e) {x | x < a or x > b}<\/p>\n

(- \u221e, a] or [a, + \u221e) {x | x < a or x > b}<\/p>\n

(- \u221e, a] or [a, + \u221e) {x | x < a or x > b}<\/p>\n

(- \u221e, a] or [a, + \u221e) {x | x < a or x > b}<\/p>\n

Properties of Absolute Values
\nIf | X | = k, then X = k or X = -k
\nIf | X | < k, then -k < X < k
\nIf | X | > k, then X > k or X < -kMetric to English (US) Conversions<\/p>\n

Distance:
\n12 in = 1 ft
\n3 ft = 1 yd 10 mm = 1 cm
\n1760 yds = 1 mi 100 cm = 1 m
\n5280 ft = 1 mi 1000m = 1 km
\n(English-Metric conversions: 1 inch = 2.54 cm; 1 mile = 1.61 km)<\/p>\n

Area:
\n144 in2 = 1 ft2 10,000 cm2 = 1 m2
\n43,560 ft2 = 1 acre 10,000 m2 = 1 hectare
\n640 acres = 1 mi2 100 hectare = 1 km2
\n(English-Metric conversions: 1 in2 = 6.45 cm2; 1 mi2 = 2.59 km2)<\/p>\n

Volume:
\n57.75 in3 = 1 qt 1 cm3 = 1 ml
\n4 qt = 1 gal 1000 ml = 1 liter
\n42 gal (petroleum) = 1 barrel 1000 liter = 1 m3
\n(English-Metric conversions: 16.39 cm3 = 1 in3; 3.79 liters = 1 gal)<\/p>\n

Mass:
\n437.5 grains = 1 oz 1000 mg = 1 g
\n16 oz = 1 lb 1000 g = 1 kg
\n2000 lb = 1 short ton 1000 kg = 1 metric ton
\n(English-Metric conversions: 453 g = 1 lb; 2.2 lb = 1 kg)<\/p>\n

Temperature:
\n(Fahrenheit – Celsius Conversions: \u00b0C = 5\/9 (\u00b0F – 32) and \u00b0F = 9\/5 \u00b0C + 32)<\/p>\n

Plane Geometry Formula<\/p>\n

Circle Square Rectangle
\nArea = \u03c0 r2 Area = s2 Area = l w
\nPerimeter = 2 \u03c0 r Perimeter = 4 s Perimeter = 2l + 2w<\/p>\n

Triangle Rhombus Trapezoid
\nArea = 1\/2 b h Area = b h Area = 1\/2 (l1 + l2) h
\nPerimeter = s1 + s2 + s3 Perimeter = 4 b Perimeter = l1 + l2 + h1 + h2
\nParallelogram Regular Polygon (n-gon)
\nArea = b h Area = (1\/2 s h) (number of sides)
\nPerimeter = 2 h1 + 2 b Perimeter = s (number of sides)
\nSolid Geometry Formula
\nCube Right Rectangular Prism Right Cylindrical Prism
\nVolume = s3 Volume = l w h Volume = \u03c0 r2 h
\nS. A. = 6 s2 S. A. = 2 l w + 2 h w + 2 l h S. A. = 2 \u03c0 r h + 2 \u03c0 r2<\/p>\n

Sphere Torus Right Triangular Prism
\nVolume = 4\/3 \u03c0 r3 Volume = 2 \u03c02 r2 R Volume = (1\/2 b h) l
\nS. A. = 4 \u03c0 r2 S. A. = 4 \u03c02 r R S. A. = b h + 2 l s + l b
\nRight Circular Cone General Cone\/Pyramid Square Pyramid
\nVolume = 1\/3 (\u03c0 r2) h Volume = 1\/3 (base area) h Volume = 1\/3 (s2) h
\nS. A. = \u03c0 r (r2 + h2)1\/2 + \u03c0 r2 S.A. = s [s + (s2 + 4h2)
\nPythagorean Theorem (Variations)<\/p>\n

For any right triangle a, b and c:<\/p>\n

a2 + b2 = c2<\/p>\n

For any non-right triangle a, b and c:<\/p>\n

a2 = b2 + c2 – 2bc cos A
\nb2 = a2 + c2 – 2ac cos B
\nc2 = a2 + b2 – 2ab cos C<\/p>\n

For any rectangular prism a, b and c, the diagonal (d) length is:<\/p>\n

d2 = a2 + b2 + c2<\/p>\n

Linear Equations
\nAn Ordered Pair: (x, y)
\nDistance between Two Ordered Pairs: d2 = \u2206 x2 + \u2206 y2 or d2 = (x2 – x1)2 + (y2 – y1)2
\nMidpoint between Two Ordered Pairs: [(x1 + x2) , (y1 + y2)]
\n2 2
\nSlope: m = \u2206 y or m = (y2 – y1) … where \u2206y = y2 – y1, \u2206 x = x2 – x1 and \u2206 x \u2260 0
\n\u2206 x (x2 – x1)
\nThe slope for Two Parallel Lines: m1 = m2
\nThe slope for Two Perpendicular Lines: m1 . m2 = -1 or m1 = -1\/m2
\nTo find the Linear Equation Using Two Ordered Pairs: (x2 – x1) m = (y2 – y1)
\nGeneral Form of a Linear Equation: Ax + By + C = 0 (A, B, C are integers, A is positive)
\nSlope Intercept Form of a Linear Equation: y = mx + b
\nConic Sections<\/p>\n

Conic Equations (Standard Form):<\/p>\n

Circle: (x – h)2 + (y – k)2 = r2 (h, k) is the center point, r is the radius from the
\ncenter to the circles (x, y) coordinates<\/p>\n

Parabolas: y – k = a(x – h)2 Parabolas, commonly written as y = ax2 + bx + c
\nx – h = a(y – k)2<\/p>\n

Ellipse: (x – h)2 + (y – k)2 = 1 (h, k) is the center point, rx is the radius length in rx2 ry2 the \u00b1 x direction, ry is the radius length in the \u00b1 y direction<\/p>\n

Hyperbola: (x – h)2 – (y – k)2 = 1 (h, k) is the center point, rx is the distance from the
\nrx2 ry2 center to the hyperbola\u2019s \u00b1 x asymptote. ry is the distance from the center to the hyperbola\u2019s \u00b1 x asymptote.Polynomials<\/p>\n

Quadratic Solutions:<\/p>\n

The solution for x from a quadratic equation ax2 + bx + c = 0, (where a \u2260 0), can be found from:<\/p>\n

Factoring:
\na2 – b2 = (a + b)(a – b) a2 + b2 … cannot be factored
\na3 – b3 = (a – b)(a2 + ab + b2) a3 + b3 = (a + b)(a2 – ab + b2)<\/p>\n

Binomial Expansions:
\n(a + b)0 = 1 (a – b)0 = 1
\n(a + b)1 = a + b (a – b)1 = a – b
\n(a + b)2 = a2 + 2ab + b2 (a – b)2 = a2 – 2ab + b2
\n(a + b)3 = a3 + 3a2b + 3ab2 + b3 (a – b)3 = a3 – 3a2b + 3ab2 – b3
\n(a + b)4 = a4 + 4a3b + 6a2b2 + 4ab3 + b4 (a – b)4 = a4 – 4a3b + 6a2b2 – 4ab3 + b4
\n(a + b)5 = a5 + 5a4b + 10a3b2 + 10a2b3 + 5ab4 + b5
\n(a – b)5 = a5 – 5a4b + 10a3b2 – 10a2b3 + 5ab4 – b5
\n(a + b)6 = a6 + 6a5b + 15a4b2 + 20a3b3 + 15a2b4 + 6ab5 + b5
\n(a – b)6 = a6 – 6a5b + 15a4b2 – 20a3b3 + 15a2b4 – 6ab5 + b5<\/p>\n

Properties of Complex Numbers<\/p>\n

( a + bi) + (c + di) = a + c + (b + d)i ( a + bi) – (c + di) = a – c + (b – d)i<\/p>\n

( a + bi) (c + di) = ac – bd + (ab + bd)i ( a + bi) (a – bi) = a2 + b2<\/p>\n

(-a)1\/2 = i(a)1\/2, a \u2265 0<\/p>\n

Properties of Exponents<\/p>\n

Properties of Rational Exponents and Radicals<\/p>\n

Basic Trigonometric Functions & Values<\/p>\n

Basic Trigonometric Ratios<\/p>\n

Sin = Opposite Cos = Adjacent Tan = Opposite
\nHypotenuse Hypotenuse Adjacent<\/p>\n

Sec = Hypotenuse Csc = Hypotenuse Cot = Adjacent
\nOpposite Adjacent Opposite<\/p>\n

Trigonometric Identities<\/p>\n

Reciprocal Identities:
\nsin \u03b8 = 1\/csc \u03b8 tan \u03b8 = 1\/cot \u03b8 cos \u03b8 = 1\/sec \u03b8
\ncsc \u03b8 = 1\/sin \u03b8 cot \u03b8 = 1\/tan \u03b8 sec \u03b8 = 1\/cos \u03b8<\/p>\n

Tangent and Cotangent Identities:
\ntan \u03b8 = sin \u03b8 \/ cos \u03b8 cot \u03b8 = cos \u03b8 \/ sin \u03b8<\/p>\n

Pythagorean Identities:
\nsin2 \u03b8 + cos2 \u03b8 =1 tan2 \u03b8 + 1 = sec2 \u03b8 1 + cot2 \u03b8 = csc2 \u03b8<\/p>\n

Double Angle Formulas:
\nsin 2\u03b8 = 2 sin \u03b8 cos \u03b8
\ncos 2\u03b8 = cos2 \u03b8 \u2212 sin2 \u03b8 cos 2\u03b8 = 2 cos2 \u03b8 \u2212 1 cos 2\u03b8 = 1\u2212 2sin2 \u03b8
\ntan 2\u03b8 = 2 tan \u03b8 \/ 1\u2212 tan2 \u03b8<\/p>\n

Sum and Difference Formulas:
\nsin (\u03b1 + \u03b2 ) = sin \u03b1 cos \u03b2 + cos \u03b1 sin \u03b2 sin (\u03b1 \u2212 \u03b2 ) = sin \u03b1 cos \u03b2 \u2212 cos \u03b1 sin \u03b2
\ncos (\u03b1 + \u03b2 ) = cos \u03b1 cos \u03b2 \u2212 sin \u03b1 sin \u03b2 cos (\u03b1 \u2212 \u03b2 ) = cos \u03b1 cos \u03b2 + sin \u03b1 sin \u03b2
\ntan(\u03b1 + \u03b2) = tan \u03b1 + tan \u03b2 \/ 1 \u2212 tan \u03b1 tan \u03b2 tan(\u03b1 \u2212 \u03b2) = tan \u03b1 \u2212 tan \u03b2 \/ 1 + tan \u03b1 tan \u03b2<\/p>\n

Graphs of Basic Trigonometric Ratios
\nTrigonometric Tables<\/p>\n

Angle Sin Cos Tan Csc Angle Sin Cos Tan Csc
\n1 0.017 1.000 0.017 57.299 46 0.719 0.695 1.036 1.390
\n2 0.035 0.999 0.035 28.654 47 0.731 0.682 1.072 1.36
\n3 0.052 0.999 0.052 19.107 48 0.743 0.669 1.111 1.346
\n4 0.070 0.998 0.070 14.336 49 0.755 0.656 1.150 1.325
\n5 0.087 0.996 0.087 11.474 50 0.766 0.643 1.192 1.305
\n6 0.105 0.995 0.105 9.567 51 0.777 0.629 1.235 1.287
\n7 0.122 0.993 0.123 8.206 52 0.788 0.616 1.280 1.269
\n8 0.139 0.990 0.141 7.185 53 0.799 0.602 1.327 1.252
\n9 0.156 0.988 0.158 6.392 54 0.809 0.588 1.376 1.236
\n10 0.174 0.985 0.176 5.759 55 0.819 0.574 1.428 1.221
\n11 0.191 0.982 0.194 5.241 56 0.829 0.559 1.483 1.206
\n12 0.208 0.978 0.213 4.810 57 0.839 0.545 1.540 1.192
\n13 0.225 0.974 0.231 4.445 58 0.848 0.530 1.600 1.179
\n14 0.242 0.970 0.249 4.134 59 0.857 0.515 1.664 1.167
\n15 0.259 0.966 0.268 3.864 60 0.866 0.500 1.732 1.155
\n16 0.276 0.961 0.287 3.628 61 0.875 0.485 1.804 1.143
\n17 0.292 0.956 0.306 3.420 62 0.883 0.469 1.881 1.133
\n18 0.309 0.951 0.325 3.236 63 0.891 0.454 1.963 1.122
\n19 0.326 0.946 0.344 3.072 64 0.899 0.438 2.050 1.113
\n20 0.342 0.940 0.364 2.924 65 0.906 0.423 2.145 1.103
\n21 0.358 0.934 0.384 2.790 66 0.914 0.407 2.246 1.095
\n22 0.375 0.927 0.404 2.669 67 0.921 0.391 2.356 1.086
\n23 0.391 0.921 0.424 2.559 68 0.927 0.375 2.475 1.079
\n24 0.407 0.914 0.445 2.459 69 0.934 0.358 2.605 1.071
\n25 0.423 0.906 0.466 2.366 70 0.940 0.342 2.747 1.064
\n26 0.438 0.899 0.488 2.281 71 0.946 0.326 2.904 1.058
\n27 0.454 0.891 0.510 2.203 72 0.951 0.309 3.078 1.051
\n28 0.469 0.883 0.532 2.130 73 0.956 0.292 3.271 1.046
\n29 0.485 0.875 0.554 2.063 74 0.961 0.276 3.487 1.040
\n30 0.500 0.866 0.577 2.000 75 0.966 0.259 3.732 1.035
\n31 0.515 0.857 0.601 1.942 76 0.970 0.242 4.011 1.031
\n32 0.530 0.848 0.625 1.887 77 0.974 0.225 4.331 1.026
\n33 0.545 0.839 0.649 1.836 78 0.978 0.208 4.705 1.022
\n34 0.559 0.829 0.675 1.788 79 0.982 0.191 5.145 1.019
\n35 0.574 0.819 0.700 1.743 80 0.985 0.174 5.671 1.015
\n36 0.588 0.809 0.727 1.701 81 0.988 0.156 6.314 1.012
\n37 0.602 0.799 0.754 1.662 82 0.990 0.139 7.115 1.010
\n38 0.616 0.788 0.781 1.624 83 0.993 0.122 8.144 1.008
\n39 0.629 0.777 0.810 1.589 84 0.995 0.105 9.514 1.006
\n40 0.643 0.766 0.839 1.556 85 0.996 0.087 11.430 1.004
\n41 0.656 0.755 0.869 1.524 86 0.998 0.070 14.301 1.002
\n42 0.669 0.743 0.900 1.494 87 0.999 0.052 19.081 1.001
\n43 0.682 0.731 0.933 1.466 88 0.999 0.035 28.636 1.001
\n44 0.695 0.719 0.966 1.440 89 1.000 0.017 57.290 1.000
\n45 0.707 0.707 1.000 1.414 90 1.000 0.000 \u00a0 1.000 Properties of Logarithmic Functions<\/p>\n

x = ay is equivalent to y = loga x ex = y is equivalent to ln y = x
\nloga (xy) = loga x + loga y loga (x\/y) = loga x – loga y loga (1\/x) = – loga x<\/p>\n

ln (xy) = ln x – ln y ln (x\/y) = ln x – ln y ln (1\/x) = – ln x<\/p>\n

loga x = log x\/ log a loga a = 1 loga 1 = 0 loga xy = y loga x<\/p>\n

loga x = ln x\/ ln a ln e = 1 ln 1 = 0 ln xy = y ln x<\/p>\n

Common Logarithm Table
\nN 0 1 2 3 4 5 6 7 8 9
\n1.0 0.0000 0.0043 0.0086 0.0128 0.0170 0.0212 0.0253 0.0294 0.0334 0.0374
\n1.1 0.0414 0.0453 0.0492 0.0531 0.0569 0.0607 0.0645 0.0682 0.0719 0.0755
\n1.2 0.0792 0.0828 0.0864 0.0899 0.0934 0.0969 0.1004 0.1038 0.1072 0.1106
\n1.3 0.1139 0.1173 0.1206 0.1239 0.1271 0.1303 0.1335 0.1367 0.1399 0.1430
\n1.4 0.1461 0.1492 0.1523 0.1553 0.1584 0.1614 0.1644 0.1673 0.1703 0.1732
\n1.5 0.1761 0.1790 0.1818 0.1847 0.1875 0.1903 0.1931 0.1959 0.1987 0.2014
\n1.6 0.2041 0.2068 0.2095 0.2122 0.2148 0.2175 0.2201 0.2227 0.2253 0.2279
\n1.7 0.2304 0.2330 0.2355 0.2380 0.2405 0.2430 0.2455 0.2480 0.2504 0.2529
\n1.8 0.2553 0.2577 0.2601 0.2625 0.2648 0.2672 0.2695 0.2718 0.2742 0.2765
\n1.9 0.2788 0.2810 0.2833 0.2856 0.2878 0.2900 0.2923 0.2945 0.2967 0.2989
\n2.0 0.3010 0.3032 0.3054 0.3075 0.3096 0.3118 0.3139 0.3160 0.3181 0.3201
\n2.1 0.3222 0.3243 0.3263 0.3284 0.3304 0.3324 0.3345 0.3365 0.3385 0.3404
\n2.2 0.3424 0.3444 0.3464 0.3483 0.3502 0.3522 0.3541 0.3560 0.3579 0.3598
\n2.3 0.3617 0.3636 0.3655 0.3674 0.3692 0.3711 0.3729 0.3747 0.3766 0.3784
\n2.4 0.3802 0.3820 0.3838 0.3856 0.3874 0.3892 0.3909 0.3927 0.3945 0.3962
\n2.5 0.3979 0.3997 0.4014 0.4031 0.4048 0.4065 0.4082 0.4099 0.4116 0.4133
\n2.6 0.4150 0.4166 0.4183 0.4200 0.4216 0.4232 0.4249 0.4265 0.4281 0.4298
\n2.7 0.4314 0.4330 0.4346 0.4362 0.4378 0.4393 0.4409 0.4425 0.4440 0.4456
\n2.8 0.4472 0.4487 0.4502 0.4518 0.4533 0.4548 0.4564 0.4579 0.4594 0.4609
\n2.9 0.4624 0.4639 0.4654 0.4669 0.4683 0.4698 0.4713 0.4728 0.4742 0.4757
\n3.0 0.4771 0.4786 0.4800 0.4814 0.4829 0.4843 0.4857 0.4871 0.4886 0.4900
\nN 0 1 2 3 4 5 6 7 8 9
\n3.1 0.4914 0.4928 0.4942 0.4955 0.4969 0.4983 0.4997 0.5011 0.5024 0.5038
\n3.2 0.5051 0.5065 0.5079 0.5092 0.5105 0.5119 0.5132 0.5145 0.5159 0.5172
\n3.3 0.5185 0.5198 0.5211 0.5224 0.5237 0.5250 0.5263 0.5276 0.5289 0.5302
\n3.4 0.5315 0.5328 0.5340 0.5353 0.5366 0.5378 0.5391 0.5403 0.5416 0.5428
\n3.5 0.5441 0.5453 0.5465 0.5478 0.5490 0.5502 0.5514 0.5527 0.5539 0.5551
\n3.6 0.5563 0.5575 0.5587 0.5599 0.5611 0.5623 0.5635 0.5647 0.5658 0.5670
\n3.7 0.5682 0.5694 0.5705 0.5717 0.5729 0.5740 0.5752 0.5763 0.5775 0.5786
\n3.8 0.5798 0.5809 0.5821 0.5832 0.5843 0.5855 0.5866 0.5877 0.5888 0.5899
\n3.9 0.5911 0.5922 0.5933 0.5944 0.5955 0.5966 0.5977 0.5988 0.5999 0.6010
\n4.0 0.6021 0.6031 0.6042 0.6053 0.6064 0.6075 0.6085 0.6096 0.6107 0.6117
\n4.1 0.6128 0.6138 0.6149 0.6160 0.6170 0.6180 0.6191 0.6201 0.6212 0.6222
\n4.2 0.6232 0.6243 0.6253 0.6263 0.6274 0.6284 0.6294 0.6304 0.6314 0.6325
\n4.3 0.6335 0.6345 0.6355 0.6365 0.6375 0.6385 0.6395 0.6405 0.6415 0.6425
\n4.4 0.6435 0.6444 0.6454 0.6464 0.6474 0.6484 0.6493 0.6503 0.6513 0.6522
\n4.5 0.6532 0.6542 0.6551 0.6561 0.6571 0.6580 0.6590 0.6599 0.6609 0.6618
\n4.6 0.6628 0.6637 0.6646 0.6656 0.6665 0.6675 0.6684 0.6693 0.6702 0.6712
\n4.7 0.6721 0.6730 0.6739 0.6749 0.6758 0.6767 0.6776 0.6785 0.6794 0.6803
\n4.8 0.6812 0.6821 0.6830 0.6839 0.6848 0.6857 0.6866 0.6875 0.6884 0.6893
\n4.9 0.6902 0.6911 0.6920 0.6928 0.6937 0.6946 0.6955 0.6964 0.6972 0.6981
\n5.0 0.6990 0.6998 0.7007 0.7016 0.7024 0.7033 0.7042 0.7050 0.7059 0.7067
\n5.1 0.7076 0.7084 0.7093 0.7101 0.7110 0.7118 0.7126 0.7135 0.7143 0.7152
\n5.2 0.7160 0.7168 0.7177 0.7185 0.7193 0.7202 0.7210 0.7218 0.7226 0.7235
\n5.3 0.7243 0.7251 0.7259 0.7267 0.7275 0.7284 0.7292 0.7300 0.7308 0.7316<\/p>\n

N 0 1 2 3 4 5 6 7 8 9
\n5.4 0.7324 0.7332 0.7340 0.7348 0.7356 0.7364 0.7372 0.7380 0.7388 0.7396
\n5.5 0.7404 0.7412 0.7419 0.7427 0.7435 0.7443 0.7451 0.7459 0.7466 0.7474
\n5.6 0.7482 0.7490 0.7497 0.7505 0.7513 0.7520 0.7528 0.7536 0.7543 0.7551
\n5.7 0.7559 0.7566 0.7574 0.7582 0.7589 0.7597 0.7604 0.7612 0.7619 0.7627
\n5.8 0.7634 0.7642 0.7649 0.7657 0.7664 0.7672 0.7679 0.7686 0.7694 0.7701
\n5.9 0.7709 0.7716 0.7723 0.7731 0.7738 0.7745 0.7752 0.7760 0.7767 0.7774
\n6.0 0.7782 0.7789 0.7796 0.7803 0.7810 0.7818 0.7825 0.7832 0.7839 0.7846
\n6.1 0.7853 0.7860 0.7868 0.7875 0.7882 0.7889 0.7896 0.7903 0.7910 0.7917
\n6.2 0.7924 0.7931 0.7938 0.7945 0.7952 0.7959 0.7966 0.7973 0.7980 0.7987
\n6.3 0.7993 0.8000 0.8007 0.8014 0.8021 0.8028 0.8035 0.8041 0.8048 0.8055
\n6.4 0.8062 0.8069 0.8075 0.8082 0.8089 0.8096 0.8102 0.8109 0.8116 0.8122
\n6.5 0.8129 0.8136 0.8142 0.8149 0.8156 0.8162 0.8169 0.8176 0.8182 0.8189
\n6.6 0.8195 0.8202 0.8209 0.8215 0.8222 0.8228 0.8235 0.8241 0.8248 0.8254
\n6.7 0.8261 0.8267 0.8274 0.8280 0.8287 0.8293 0.8299 0.8306 0.8312 0.8319
\n6.8 0.8325 0.8331 0.8338 0.8344 0.8351 0.8357 0.8363 0.8370 0.8376 0.8382
\n6.9 0.8388 0.8395 0.8401 0.8407 0.8414 0.8420 0.8426 0.8432 0.8439 0.8445
\n7.0 0.8451 0.8457 0.8463 0.8470 0.8476 0.8482 0.8488 0.8494 0.8500 0.8506
\n7.1 0.8513 0.8519 0.8525 0.8531 0.8537 0.8543 0.8549 0.8555 0.8561 0.8567
\n7.2 0.8573 0.8579 0.8585 0.8591 0.8597 0.8603 0.8609 0.8615 0.8621 0.8627
\n7.3 0.8633 0.8639 0.8645 0.8651 0.8657 0.8663 0.8669 0.8675 0.8681 0.8686
\n7.4 0.8692 0.8698 0.8704 0.8710 0.8716 0.8722 0.8727 0.8733 0.8739 0.8745
\n7.5 0.8751 0.8756 0.8762 0.8768 0.8774 0.8779 0.8785 0.8791 0.8797 0.8802
\n7.6 0.8808 0.8814 0.8820 0.8825 0.8831 0.8837 0.8842 0.8848 0.8854 0.8859
\nN 0 1 2 3 4 5 6 7 8 9
\n7.7 0.8865 0.8871 0.8876 0.8882 0.8887 0.8893 0.8899 0.8904 0.8910 0.8915
\n7.8 0.8921 0.8927 0.8932 0.8938 0.8943 0.8949 0.8954 0.8960 0.8965 0.8971
\n7.9 0.8976 0.8982 0.8987 0.8993 0.8998 0.9004 0.9009 0.9015 0.9020 0.9025
\n8.0 0.9031 0.9036 0.9042 0.9047 0.9053 0.9058 0.9063 0.9069 0.9074 0.9079
\n8.1 0.9085 0.9090 0.9096 0.9101 0.9106 0.9112 0.9117 0.9122 0.9128 0.9133
\n8.2 0.9138 0.9143 0.9149 0.9154 0.9159 0.9165 0.9170 0.9175 0.9180 0.9186
\n8.3 0.9191 0.9196 0.9201 0.9206 0.9212 0.9217 0.9222 0.9227 0.9232 0.9238
\n8.4 0.9243 0.9248 0.9253 0.9258 0.9263 0.9269 0.9274 0.9279 0.9284 0.9289
\n8.5 0.9294 0.9299 0.9304 0.9309 0.9315 0.9320 0.9325 0.9330 0.9335 0.9340
\n8.6 0.9345 0.9350 0.9355 0.9360 0.9365 0.9370 0.9375 0.9380 0.9385 0.9390
\n8.7 0.9395 0.9400 0.9405 0.9410 0.9415 0.9420 0.9425 0.9430 0.9435 0.9440
\n8.8 0.9445 0.9450 0.9455 0.9460 0.9465 0.9469 0.9474 0.9479 0.9484 0.9489
\n8.9 0.9494 0.9499 0.9504 0.9509 0.9513 0.9518 0.9523 0.9528 0.9533 0.9538
\n9.0 0.9542 0.9547 0.9552 0.9557 0.9562 0.9566 0.9571 0.9576 0.9581 0.9586
\n9.1 0.9590 0.9595 0.9600 0.9605 0.9609 0.9614 0.9619 0.9624 0.9628 0.9633
\n9.2 0.9638 0.9643 0.9647 0.9652 0.9657 0.9661 0.9666 0.9671 0.9675 0.9680
\n9.3 0.9685 0.9689 0.9694 0.9699 0.9703 0.9708 0.9713 0.9717 0.9722 0.9727
\n9.4 0.9731 0.9736 0.9741 0.9745 0.9750 0.9754 0.9759 0.9763 0.9768 0.9773
\n9.5 0.9777 0.9782 0.9786 0.9791 0.9795 0.9800 0.9805 0.9809 0.9814 0.9818
\n9.6 0.9823 0.9827 0.9832 0.9836 0.9841 0.9845 0.9850 0.9854 0.9859 0.9863
\n9.7 0.9868 0.9872 0.9877 0.9881 0.9886 0.9890 0.9894 0.9899 0.9903 0.9908
\n9.8 0.9912 0.9917 0.9921 0.9926 0.9930 0.9934 0.9939 0.9943 0.9948 0.9952
\n9.9 0.9956 0.9961 0.9965 0.9969 0.9974 0.9978 0.9983 0.9987 0.9991 0.9996<\/p>\n","protected":false},"author":14,"menu_order":2,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"back-matter-type":[],"contributor":[],"license":[],"class_list":["post-407","back-matter","type-back-matter","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/pressbooks\/v2\/back-matter\/407","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/pressbooks\/v2\/back-matter"}],"about":[{"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/wp\/v2\/types\/back-matter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/wp\/v2\/users\/14"}],"version-history":[{"count":2,"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/pressbooks\/v2\/back-matter\/407\/revisions"}],"predecessor-version":[{"id":409,"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/pressbooks\/v2\/back-matter\/407\/revisions\/409"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/pressbooks\/v2\/back-matter\/407\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/wp\/v2\/media?parent=407"}],"wp:term":[{"taxonomy":"back-matter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/pressbooks\/v2\/back-matter-type?post=407"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/wp\/v2\/contributor?post=407"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/wp\/v2\/license?post=407"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}