Exponential and Logarithmic Functions

# Evaluate and Graph Exponential Functions

### Learning Objectives

By the end of this section, you will be able to:

• Graph exponential functions
• Solve Exponential equations
• Use exponential models in applications

Before you get started, take this readiness quiz.

1. Simplify:

If you missed this problem, review (Figure).

2. Evaluate:

If you missed this problem, review (Figure).

3. Evaluate:

If you missed this problem, review (Figure).

### Graph Exponential Functions

The functions we have studied so far do not give us a model for many naturally occurring phenomena. From the growth of populations and the spread of viruses to radioactive decay and compounding interest, the models are very different from what we have studied so far. These models involve exponential functions.

An exponential function is a function of the form where and

Exponential Function

An exponential function, where and is a function of the form

Notice that in this function, the variable is the exponent. In our functions so far, the variables were the base.

Our definition says If we let then becomes Since for all real numbers, This is the constant function.

Our definition also says If we let a base be negative, say then is not a real number when

In fact, would not be a real number any time is a fraction with an even denominator. So our definition requires

By graphing a few exponential functions, we will be able to see their unique properties.

On the same coordinate system graph and

We will use point plotting to graph the functions.

Graph:

Graph:

If we look at the graphs from the previous Example and Try Its, we can identify some of the properties of exponential functions.

The graphs of and as well as the graphs of and all have the same basic shape. This is the shape we expect from an exponential function where

We notice, that for each function, the graph contains the point This make sense because for any a.

The graph of each function, also contains the point The graph of contained and the graph of contained This makes sense as

Notice too, the graph of each function also contains the point The graph of contained and the graph of contained This makes sense as

What is the domain for each function? From the graphs we can see that the domain is the set of all real numbers. There is no restriction on the domain. We write the domain in interval notation as

Look at each graph. What is the range of the function? The graph never hits the -axis. The range is all positive numbers. We write the range in interval notation as

Whenever a graph of a function approaches a line but never touches it, we call that line an asymptote. For the exponential functions we are looking at, the graph approaches the -axis very closely but will never cross it, we call the line the x-axis, a horizontal asymptote.

Properties of the Graph of when
 Domain Range x-intercept None y-intercept Contains Asymptote -axis, the line

Our definition of an exponential function says but the examples and discussion so far has been about functions where What happens when ? The next example will explore this possibility.

On the same coordinate system, graph and

We will use point plotting to graph the functions.

Graph:

Graph:

Now let’s look at the graphs from the previous Example and Try Its so we can now identify some of the properties of exponential functions where

The graphs of and as well as the graphs of and all have the same basic shape. While this is the shape we expect from an exponential function where the graphs go down from left to right while the previous graphs, when went from up from left to right.

We notice that for each function, the graph still contains the point (0, 1). This make sense because for any a.

As before, the graph of each function, also contains the point The graph of contained and the graph of contained This makes sense as

Notice too that the graph of each function, also contains the point The graph of contained and the graph of contained This makes sense as

What is the domain and range for each function? From the graphs we can see that the domain is the set of all real numbers and we write the domain in interval notation as Again, the graph never hits the -axis. The range is all positive numbers. We write the range in interval notation as

We will summarize these properties in the chart below. Which also include when

Properties of the Graph of
when when
Domain Domain
Range Range
-intercept none -intercept none
-intercept -intercept
Contains Contains
Asymptote -axis, the line Asymptote -axis, the line
Basic shape increasing Basic shape decreasing

It is important for us to notice that both of these graphs are one-to-one, as they both pass the horizontal line test. This means the exponential function will have an inverse. We will look at this later.

When we graphed quadratic functions, we were able to graph using translation rather than just plotting points. Will that work in graphing exponential functions?

On the same coordinate system graph and

We will use point plotting to graph the functions.

On the same coordinate system, graph: and

On the same coordinate system, graph: and

Looking at the graphs of the functions and in the last example, we see that adding one in the exponent caused a horizontal shift of one unit to the left. Recognizing this pattern allows us to graph other functions with the same pattern by translation.

Let’s now consider another situation that might be graphed more easily by translation, once we recognize the pattern.

On the same coordinate system graph and

We will use point plotting to graph the functions.

On the same coordinate system, graph: and

On the same coordinate system, graph: and

Looking at the graphs of the functions and in the last example, we see that subtracting 2 caused a vertical shift of down two units. Notice that the horizontal asymptote also shifted down 2 units. Recognizing this pattern allows us to graph other functions with the same pattern by translation.

All of our exponential functions have had either an integer or a rational number as the base. We will now look at an exponential function with an irrational number as the base.

Before we can look at this exponential function, we need to define the irrational number, e. This number is used as a base in many applications in the sciences and business that are modeled by exponential functions. The number is defined as the value of as n gets larger and larger. We say, as n approaches infinity, or increases without bound. The table shows the value of for several values of

1 2
2 2.25
5 2.48832
10 2.59374246
100 2.704813829…
1,000 2.716923932…
10,000 2.718145927…
100,000 2.718268237…
1,000,000 2.718280469…
1,000,000,000 2.718281827…

The number e is like the number in that we use a symbol to represent it because its decimal representation never stops or repeats. The irrational number e is called the natural base.

Natural Base

The number e is defined as the value of as n increases without bound. We say, as n approaches infinity,

The exponential function whose base is is called the natural exponential function.

Natural Exponential Function

The natural exponential function is an exponential function whose base is

The domain is and the range is

Let’s graph the function on the same coordinate system as and

Notice that the graph of is “between” the graphs of and Does this make sense as ?

### Solve Exponential Equations

Equations that include an exponential expression are called exponential equations. To solve them we use a property that says as long as and if then it is true that In other words, in an exponential equation, if the bases are equal then the exponents are equal.

One-to-One Property of Exponential Equations

For and

To use this property, we must be certain that both sides of the equation are written with the same base.

How to Solve an Exponential Equation

Solve:

Solve:

Solve:

The steps are summarized below.

How to Solve an Exponential Equation
1. Write both sides of the equation with the same base, if possible.
2. Write a new equation by setting the exponents equal.
3. Solve the equation.
4. Check the solution.

In the next example, we will use our properties on exponents.

Solve .

 Use the Property of Exponents: Write a new equation by setting the exponents equal. Solve the equation. Check the solutions.

Solve:

Solve:

### Use Exponential Models in Applications

Exponential functions model many situations. If you own a bank account, you have experienced the use of an exponential function. There are two formulas that are used to determine the balance in the account when interest is earned. If a principal, P, is invested at an interest rate, r, for t years, the new balance, A, will depend on how often the interest is compounded. If the interest is compounded n times a year we use the formula If the interest is compounded continuously, we use the formula These are the formulas for compound interest.

Compound Interest

For a principal, P, invested at an interest rate, r, for t years, the new balance, A, is:

*** QuickLaTeX cannot compile formula:
\begin{array}{}\\ \\ A=P{\left(1+\frac{r}{n}\right)}^{nt}\hfill & & & & & \text{when compounded}\phantom{\rule{0.2em}{0ex}}n\phantom{\rule{0.2em}{0ex}}\text{times a year.}\hfill \\ A=P{e}^{rt}\hfill & & & & & \text{when compounded continuously.}\hfill \end{array}

*** Error message:
Missing # inserted in alignment preamble.
leading text: $\begin{array}{} Missing$ inserted.
leading text: $\begin{array}{}\\ \\ A=P{\left Extra }, or forgotten$.
leading text: ...ray}{}\\ \\ A=P{\left(1+\frac{r}{n}\right)}
Missing } inserted.
leading text: ...=P{\left(1+\frac{r}{n}\right)}^{nt}\hfill &
Extra }, or forgotten $. leading text: ...=P{\left(1+\frac{r}{n}\right)}^{nt}\hfill & Missing } inserted. leading text: ...=P{\left(1+\frac{r}{n}\right)}^{nt}\hfill & Extra }, or forgotten$.
leading text: ...=P{\left(1+\frac{r}{n}\right)}^{nt}\hfill &
Missing } inserted.
leading text: ...=P{\left(1+\frac{r}{n}\right)}^{nt}\hfill &
Extra }, or forgotten . leading text: ...=P{\left(1+\frac{r}{n}\right)}^{nt}\hfill & Missing } inserted. leading text: ...=P{\left(1+\frac{r}{n}\right)}^{nt}\hfill &  As you work with the Interest formulas, it is often helpful to identify the values of the variables first and then substitute them into the formula. A total of was invested in a college fund for a new grandchild. If the interest rate is *** QuickLaTeX cannot compile formula: 5\text{%}, *** Error message: File ended while scanning use of \text@. Emergency stop.  how much will be in the account in 18 years by each method of compounding? compound quarterly compound monthly compound continuously Angela invested in a savings account. If the interest rate is how much will be in the account in 10 years by each method of compounding? compound quarterly compound monthly compound continuously Allan invested ?10,000 in a mutual fund. If the interest rate is *** QuickLaTeX cannot compile formula: 5\text{%}, *** Error message: File ended while scanning use of \text@. Emergency stop.  how much will be in the account in 15 years by each method of compounding? compound quarterly compound monthly compound continuously ?21,071.81 ?21,137.04 ?21,170.00 Other topics that are modeled by exponential functions involve growth and decay. Both also use the formula we used for the growth of money. For growth and decay, generally we use as the original amount instead of calling it the principal. We see that exponential growth has a positive rate of growth and exponential decay has a negative rate of growth. Exponential Growth and Decay For an original amount, that grows or decays at a rate, r, for a certain time, t, the final amount, A, is: Exponential growth is typically seen in the growth of populations of humans or animals or bacteria. Our next example looks at the growth of a virus. Chris is a researcher at the Center for Disease Control and Prevention and he is trying to understand the behavior of a new and dangerous virus. He starts his experiment with 100 of the virus that grows at a rate of 25% per hour. He will check on the virus in 24 hours. How many viruses will he find? Another researcher at the Center for Disease Control and Prevention, Lisa, is studying the growth of a bacteria. She starts his experiment with 50 of the bacteria that grows at a rate of *** QuickLaTeX cannot compile formula: 15\text{%} *** Error message: File ended while scanning use of \text@. Emergency stop.  per hour. He will check on the bacteria every 8 hours. How many bacteria will he find in 8 hours? She will find 166 bacteria. Maria, a biologist is observing the growth pattern of a virus. She starts with 100 of the virus that grows at a rate of per hour. She will check on the virus in 24 hours. How many viruses will she find? She will find 1,102 viruses. Access these online resources for additional instruction and practice with evaluating and graphing exponential functions. ### Key Concepts • Properties of the Graph of when when Domain Domain Range Range -intercept none -intercept none -intercept -intercept Contains Contains Asymptote -axis, the line Asymptote -axis, the line Basic shape increasing Basic shape decreasing • One-to-One Property of Exponential Equations: For and • How to Solve an Exponential Equation 1. Write both sides of the equation with the same base, if possible. 2. Write a new equation by setting the exponents equal. 3. Solve the equation. 4. Check the solution. • Compound Interest: For a principal, invested at an interest rate, for years, the new balance, is *** QuickLaTeX cannot compile formula: \begin{array}{}\\ \\ A=P{\left(1+\frac{r}{n}\right)}^{nt}\hfill & & & & & \text{when compounded}\phantom{\rule{0.2em}{0ex}}n\phantom{\rule{0.2em}{0ex}}\text{times a year.}\hfill \\ A=P{e}^{rt}\hfill & & & & & \text{when compounded continuously.}\hfill \end{array} *** Error message: Missing # inserted in alignment preamble. leading text:\begin{array}{}
Missing $inserted. leading text:$\begin{array}{}\\ \\ A=P{\left
Extra }, or forgotten $. leading text: ...ray}{}\\ \\ A=P{\left(1+\frac{r}{n}\right)} Missing } inserted. leading text: ...=P{\left(1+\frac{r}{n}\right)}^{nt}\hfill & Extra }, or forgotten$.
leading text: ...=P{\left(1+\frac{r}{n}\right)}^{nt}\hfill &
Missing } inserted.
leading text: ...=P{\left(1+\frac{r}{n}\right)}^{nt}\hfill &
Extra }, or forgotten $. leading text: ...=P{\left(1+\frac{r}{n}\right)}^{nt}\hfill & Missing } inserted. leading text: ...=P{\left(1+\frac{r}{n}\right)}^{nt}\hfill & Extra }, or forgotten$.
leading text: ...=P{\left(1+\frac{r}{n}\right)}^{nt}\hfill &
Missing } inserted.
leading text: ...=P{\left(1+\frac{r}{n}\right)}^{nt}\hfill &


• Exponential Growth and Decay: For an original amount, that grows or decays at a rate, for a certain time the final amount, is

#### Practice Makes Perfect

Graph Exponential Functions

In the following exercises, graph each exponential function.

In the following exercises, graph each function in the same coordinate system.

In the following exercises, graph each exponential function.

Solve Exponential Equations

In the following exercises, solve each equation.

In the following exercises, match the graphs to one of the following functions:

Use exponential models in applications

In the following exercises, use an exponential model to solve.

Edgar accumulated in credit card debt. If the interest rate is per year, and he does not make any payments for 2 years, how much will he owe on this debt in 2 years by each method of compounding?

compound quarterly

compound monthly

compound continuously

Cynthia invested in a savings account. If the interest rate is how much will be in the account in 10 years by each method of compounding?

compound quarterly

compound monthly

compound continuously

Rochelle deposits in an IRA. What will be the value of her investment in 25 years if the investment is earning

*** QuickLaTeX cannot compile formula:
8\text{%}

*** Error message:
File ended while scanning use of \text@.
Emergency stop.



per year and is compounded continuously?

Nazerhy deposits in a certificate of deposit. The annual interest rate is and the interest will be compounded quarterly. How much will the certificate be worth in 10 years?

A researcher at the Center for Disease Control and Prevention is studying the growth of a bacteria. He starts his experiment with 100 of the bacteria that grows at a rate of per hour. He will check on the bacteria every 8 hours. How many bacteria will he find in 8 hours?

223 bacteria

A biologist is observing the growth pattern of a virus. She starts with 50 of the virus that grows at a rate of per hour. She will check on the virus in 24 hours. How many viruses will she find?

In the last ten years the population of Indonesia has grown at a rate of

*** QuickLaTeX cannot compile formula:
1.12\text{%}

*** Error message:
File ended while scanning use of \text@.
Emergency stop.



per year to 258,316,051. If this rate continues, what will be the population in 10 more years?

288,929,825

In the last ten years the population of Brazil has grown at a rate of

*** QuickLaTeX cannot compile formula:
0.9\text{%}

*** Error message:
File ended while scanning use of \text@.
Emergency stop.



per year to 205,823,665. If this rate continues, what will be the population in 10 more years?

#### Writing Exercises

Explain how you can distinguish between exponential functions and polynomial functions.

Answers will vary.

Compare and contrast the graphs of and .

What happens to an exponential function as the values of decreases? Will the graph ever cross the

-axis? Explain.

Answers will vary.

#### Self Check

After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

After reviewing this checklist, what will you do to become confident for all objectives?

### Glossary

asymptote
A line which a graph of a function approaches closely but never touches.
exponential function
An exponential function, where and is a function of the form
natural base
The number e is defined as the value of as n gets larger and larger. We say, as n increases without bound,
natural exponential function
The natural exponential function is an exponential function whose base is e: The domain is and the range is

## License

Intermediate Algebra Copyright © 2017 by OSCRiceUniversity is licensed under a Creative Commons Attribution 4.0 International License, except where otherwise noted.