1. 5 Exponential and Logarithmic Functions
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2. What You Should Learn
Recognize and evaluate exponential functions with base a.
Graph exponential functions and use the One-to-One Property.
Recognize, evaluate, and graph exponential functions with base e.

3. Exponential Functions

4. Exponential Functions

5. Graphs of Exponential Functions

6. Example 2 – Graphs of y = ax
In the same coordinate plane, sketch the graph of each function.
a. f (x) = 2x
b. g (x) = 4x

7. Example 2 – Solution
The table below lists some values for each function, and Figure 5.1 shows the graphs of the two functions.
Note that both graphs are increasing. Moreover, the graph of g(x) = 4x is increasing more rapidly than the graph of f (x) = 2x.
Figure 5.1

8. Graphs of Exponential Functions
The basic characteristics of exponential functions y = ax and y = a–x are summarized in Figures 5.3 and 5.4.
Graph of y = ax, a > 1
Domain: ( , )
Range: (0, )
y-intercept: (0, 1)
Increasing
x-axis is a horizontal asymptote (ax → 0 as x→ ).
Continuous
Figure 5.3

9. Graphs of Exponential Functions
Graph of y = a–x, a > 1
Domain: ( , )
Range: (0, )
y-intercept: (0, 1)
Decreasing
x-axis is a horizontal asymptote (a–x → 0 as x→ ).
Continuous
From Figures 5.3 and 5.4, you can see that the graph of an exponential function is always increasing or always decreasing.
Figure 5.4

10. Graphs of Exponential Functions
As a result, the graphs pass the Horizontal Line Test, and therefore the functions are one-to-one functions.
You can use the following One-to-One Property to solve simple exponential equations.
For a > 0 and a ≠ 1, ax = ay if and only if x = y.
One-to-One Property

11. The Natural Base e

12. The Natural Base e
In many applications, the most convenient choice for a base is the irrational number
e 
This number is called the natural base.
The function given by f (x) = ex is
called the natural exponential
function. Its graph is shown in
Figure 5.9.
Figure 5.9

13. The Natural Base e Be sure you see that for the exponential function
f (x) = ex, e is the constant , whereas x
is the variable.

14. Example 6 – Evaluating the Natural Exponential Function
Use a calculator to evaluate the function given by
f (x) = ex at each indicated value of x.
a. x = –2
b. x = –1
c. x = 0.25
d. x = –0.3

15. Example 6 – Solution a. f (–2) = e–2 0.1353353
Function Value Graphing Calculator Keystrokes Display
a. f (–2) = e–
b. f (–1) = e–
c. f (0.25) = e
d. f (–0.3) = e–

16. What You Should Learn
Recognize and evaluate logarithmic functions with base a.
Graph logarithmic functions.
Recognize, evaluate, and graph natural logarithmic functions.

17. Logarithmic Functions

18. Logarithmic Functions

19. Logarithmic Functions
The equations
y = loga x and x = ay
are equivalent. The first equation is in logarithmic form and the second is in exponential form.
For example, the logarithmic equation 2 = log3 9 can be rewritten in exponential form as 9 = 32. The exponential equation 53 = 125 can be rewritten in logarithmic form as log5 125 = 3.

20. Logarithmic Functions
When evaluating logarithms, remember that a logarithm is an exponent. This means that loga x is the exponent to which a must be raised to obtain x.
For instance, log2 8 = 3 because 2 must be raised to the third power to get 8.

21. Example 1 – Evaluating Logarithms
Use the definition of logarithmic function to evaluate each logarithm at the indicated value of x.
a. f (x) = log2 x, x = b. f (x) = log3 x, x = 1
c. f (x) = log4 x, x = d. f (x) = log10 x, x =
Solution:
a. f (32) = log because 25 = 32.
= 5
b. f (1) = log because 30 = 1.
= 0

22. Example 1 – Solution c. f (2) = log4 2 because 41/2 = = 2. =
cont’d
c. f (2) = log because 41/2 = = 2. =
d because

23. Logarithmic Functions
The logarithmic function with base 10 is called the common logarithmic function.
It is denoted by log10 or simply by log.
On most calculators, this function is denoted by
Properties of the logarithmic function with base a:

24. Graphs of Logarithmic Functions

25. Example 5 – Graphs of Exponential and Logarithmic Functions
In the same coordinate plane, sketch the graph of each function.
a. f (x) = 2x
b. g(x) = log2 x

26. Example 5(a) – Solution For f (x) = 2x, construct a table of values.
g(x) = log2 x is the inverse function
of f (x) = 2x
Figure 5.14

27. Graphs of Logarithmic Functions
The basic characteristics of Graph of y = loga x, a  1
• Domain: (0, )
• Range: ( , )
• x-intercept: (1, 0)
• Increasing
• One-to-one, therefore has an inverse function
• y-axis is a vertical asymptote (loga x → as x → 0+).
• Continuous
• Reflection of graph of y = ax about the line y = x.

28. The Natural Logarithmic Function

29. The Natural Logarithmic Function
Natural logarithmic function and is denoted by the special symbol ln x, read as “the natural log of x” or “el en of x.” Note that the natural logarithm is written without a base. The base is understood to be e.

30. The Natural Logarithmic Function
f (x) = ex and g(x) = In x are inverse functions of each other.
On most calculators, the natural logarithm is denoted by
Reflection of graph of f (x) = ex about the line y = x.
Figure 5.19

31. Example 8 – Evaluating the Natural Logarithmic Function
Use a calculator to evaluate the function given by f (x) = In x for each value of x.
a. x = 2
b. x = 0.3
c. x = –1
d. x = 1 +

32. Example 8 – Solution Function Value Graphing Calculator Display
Keystrokes

33. The Natural Logarithmic Function

34. What You Should Learn
Use the change-of-base formula to rewrite and evaluate logarithmic expressions.
Use properties of logarithms to evaluate or rewrite logarithmic expressions.
Use properties of logarithms to expand or condense logarithmic expressions.

35. Change of Base

36. Change of Base
Most calculators have only two types of log keys, one for common logarithms (base 10) and one for natural logarithms (base e).
Evaluate logarithms with other bases by using the following change-of-base formula.

37. Example 1 – Changing Bases Using Common Logarithms
Use a calculator.
Simplify.

38. Properties of Logarithms

39. Properties of Logarithms

40. Example 3 – Using Properties of Logarithms
Write each logarithm in terms of ln 2 and ln 3.
a. ln 6 b. ln
Solution:
a. ln 6 = ln (2  3)
= ln 2 + ln 3
b. ln = ln 2 – ln 27
= ln 2 – ln 33
= ln 2 – 3 ln 3
Rewrite 6 as 2  3.
Product Property
Quotient Property
Rewrite 27 as 33.
Power Property

41. Rewriting Logarithmic Expressions

42. Example 5 – Expanding Logarithmic Expressions
Expand each logarithmic expression.
a. log4 5x3y b.
Solution:
a. log4 5x3y = log4 5 + log4 x3 + log4 y
= log log4 x + log4 y
Product Property
Power Property

43. Example 5 – Solution b. cont’d Rewrite using rational exponent.
Quotient Property
Power Property

44. What You Should Learn
Solve exponential and logarithmic equations.

45. Solving Exponential Equations

46. Example 2 – Solving Exponential Equations
Solve each equation and approximate the result to three decimal places, if necessary.
a. e – x2 = e – 3x – 4
b. 3(2x) = 42

47. Example 2(a) – Solution e –x2 = e – 3x – 4 –x2 = –3x – 4
The solutions are x = –1 and x = 4. Check these in the original equation.
Write original equation.
One-to-One Property
Write in general form.
Factor.
x = –1
Set 1st factor equal to 0.
x = 4
Set 2nd factor equal to 0.

48. Example 2(b) – Solution 3(2x) = 42 2x = 14 log2 2x = log2 14
cont’d
3(2x) = 42
2x = 14
log2 2x = log2 14
x = log2 14
x =
 3.807
The solution is x = log2 14  Check this in the original equation.
Write original equation.
Divide each side by 3.
Take log (base 2) of each side.
Inverse Property
Change-of-base formula

49. Solving Logarithmic Equations

50. Solving Logarithmic Equations
To solve a logarithmic equation, you can write it in exponential form.
ln x = 3
eln x = e3
x = e3
This procedure is called exponentiating each side of an equation.
Logarithmic form
Exponentiate each side.
Exponential form

51. Example 6 – Solving Logarithmic Equations
a. ln x = 2
eln x = e2
x = e2
b. log3(5x – 1) = log3(x + 7)
5x – 1 = x + 7
4x = 8
x = 2
Original equation
Exponentiate each side.
Inverse Property
Original equation
One-to-One Property
Add –x and 1 to each side.
Divide each side by 4.

52. Example 6 – Solving Logarithmic Equations
cont’d
c. log6(3x + 14) – log6 5 = log6 2x
3x + 14 = 10x
–7x = –14
x = 2
Original equation
Quotient Property of Logarithms
One-to-One Property
Cross multiply.
Isolate x.
Divide each side by –7.