1. Copyright © Cengage Learning. All rights reserved.
Chapter 9
Exponential and Logarithmic Functions
Copyright © Cengage Learning. All rights reserved.

2. Copyright © Cengage Learning. All rights reserved.
9.1 Exponential Functions
Copyright © Cengage Learning. All rights reserved.

3. What You Should Learn Evaluate exponential functions.
Graph exponential functions.
Evaluate the natural base e and graph natural exponential functions.
Use exponential functions to solve application problems.

4. Exponential Functions

5. Exponential Functions
In this section, you will study a new type of function called an exponential function. Whereas polynomial and rational functions have terms with variable bases and constant exponents, exponential functions have terms with constant bases and variable exponents.
Here are some examples.

6. Exponential Functions

7. Exponential Functions
The base a = 1 is excluded because f(x) = 1x = 1 is a constant function, not an exponential function.
You have learned to evaluate ax for integer and rational values of x.
For example, you know that
However, to evaluate ax for any real number x, you need to interpret forms with irrational exponents, such as or

8. Exponential Functions
For the purposes of this section, it is sufficient to think of a number such as , where as the number that has the successively closer approximations
a1.4, a1.41, a1.414, a1.4142, a , a ,
The rules of exponents can be extended to cover exponential functions.

9. Exponential Functions

10. Exponential Functions
To evaluate exponential functions with a calculator, you can use the exponential key or For example, to evaluate 3–1.3, you can use the following keystrokes.

11. Example 1 – Evaluating Exponential Functions
Evaluate each function. Use a calculator only if it is necessary or more efficient.
Function Values
a. f(x) = 2x x = 3, x = –4, x = 
b. g(x) = 12x x = 3, x = –0.1, x =
c. h(x) = (1.04)2x x = 0, x = –2, x =

12. Example 1 – Solution Evaluation Comment a. f(3) = 23 = 8 f(–4) = 2–4
b. g(3) = 123
g(–0.1) = 12–0.1
g = 125/7
Calculator is not necessary. =
Calculator is not necessary.
≈ 8.825
Calculator is necessary.
= 1728
Calculator is more efficient.
≈ 0.780
Calculator is necessary.
≈ 5.900
Calculator is necessary.

13. Example 1 – Solution c. h(0) = (1.04)20 Evaluation Comment = (1.04)0
cont’d
Evaluation Comment
c. h(0) = (1.04)20
h(–2) = (1.04)2(–2)
h( ) = (1.04)2
= (1.04)0
= 1
Calculator is not necessary.
≈ 0.855
Calculator is more efficient.
≈ 1.117
Calculator is necessary.

14. Graphs of Exponential Functions

15. Example 2 – The Graphs of Exponential Functions
In the same coordinate plane, sketch the graph of each function. Determine the domain and range.
a. f(x) = 2x
b. g(x) = 4x
Solution:
The table lists some values of each function.

16. Example 2 – Solution Figure 9.1 shows the graph of each function.
cont’d
Figure 9.1 shows the graph of each function.
From the graphs, you can see that the domain of each function is the set of all real numbers and that the range of each function is the set of all positive real numbers.
Figure 9.1

17. Graphs of Exponential Functions
For a > 1, the values of the function y = ax increase as x increases and the values of the function y = a–x = (1/a)x decrease as x increases.

18. Graphs of Exponential Functions
The graphs shown in Figure 9.3 are typical of the graphs of exponential functions.
Note that each graph has a y-intercept at (0, 1) and a horizontal asymptote of y = 0 (the x-axis).
Figure 9.3

19. The Natural Exponential Function

20. The Natural Exponential Function
So far, integers or rational numbers have been used as bases of exponential functions. In many applications of exponential functions, the convenient choice for a base is the following irrational number, denoted by the letter “e.”
e ≈ Natural base
This number is called the natural base. The function
f(x) = ex Natural exponential function
is called the natural exponential function.

21. The Natural Exponential Function
To evaluate the natural exponential function, you need a calculator, preferably one having a natural exponential key Here are some examples of how to use such a calculator to evaluate the natural exponential function.

22. The Natural Exponential Function
When evaluating the natural exponential function, remember that e is the constant number , and x is a variable.
The following table displays the function evaluated at several values.

23. The Natural Exponential Function
You can sketch a graph for the values displayed in the table as shown in Figure 9.8.
From the graph, notice the following characteristics of the natural exponential function.
• Domain: (– , )
• Range: (0, )
• Intercept: (0, 1)
• Increasing (moves up to the right)
• Asymptote: x-axis
Figure 9.8

24. Applications

25. Example 6 – Radioactive Decay
A particular radioactive element has a half-life of 25 years. For an initial mass of 10 grams, the mass y (in grams) that remains after t years is given by
How much of the initial mass remains after 120 years?

26. Example 6 – Solution When t = 120, the mass is given by
So, after 120 years, the mass has decayed from an initial amount of 10 grams to only gram.

27. Example 6 – Solution
cont’d
Note in Figure 9.9 that the graph of the function shows the 25-year half-life. That is, after 25 years the mass is grams (half of the original), after another 25 years the mass is 2.5 grams, and so on.
Figure 9.9

28. Applications
One of the most familiar uses of exponential functions involves compound interest. For instance, a principal P is invested at an annual interest rate r (in decimal form), compounded once a year. If the interest is added to the principal at the end of the year, the balance is
A = P + Pr
= P(1 + r).

29. Applications
This pattern of multiplying the previous principal by (1 + r) is then repeated each successive year, as shown below.
Time in Years Balance at Given Time
0 A = P
1 A = P(1 + r)
2 A = P(1 + r)(1 + r) = P(1 + r)2
3 A = P(1 + r)2(1 + r) = P(1 + r)3
t A = P(1 + r)t

30. Applications
To account for more frequent compounding of interest (such as quarterly or monthly compounding), let n be the number of compoundings per year and let t be the number of years.
Then the rate per compounding is r/n and the account balance after t years is

31. Applications
A second method that banks use to compute interest is called continuous compounding. The formula for the balance for this type of compounding is
A = Pert.
The formulas for both types of compounding are summarized as