1. Chapter 5: Inverse, Exponential, and Logarithmic Functions
5.1 Inverse Functions 5.2 Exponential Functions 5.3 Logarithms and Their Properties 5.4 Logarithmic Functions 5.5 Exponential and Logarithmic Equations and Inequalities 5.6 Further Applications and Modeling with Exponential and Logarithmic Functions

2. Logarithms and Their Properties
A logarithm is an exponent, and loga x is the exponent to which a must be raised in order to obtain x. The number a is called the base of the logarithm, and x is called the argument of the expression loga x. The value of x must always be positive.

3. Examples of Logarithms
Fill in the blanks above

4. Examples of Logarithms

5. Solving Logarithmic Equations

6. Solving Logarithmic Equations

7. The Common Logarithm – Base 10
Solution Use a calculator.

8. The Common Logarithm – Base 10
Solution Use a calculator.

9. Application of the Common Logarithm
a) Find the pH of a solution with [H3O+] = 2.5×10-4. b) Find the hydronium ion concentration of a solution with pH = 7.1.

10. Application of the Common Logarithm
a) Find the pH of a solution with [H3O+] = 2.5×10-4. b) Find the hydronium ion concentration of a solution with pH = 7.1. Solution a) pH = –log [H3O+] = –log [2.5×10-4]  3.6 b) 7.1 = –log [H3O+]  –7.1 = log [H3O+]  [H3O+] =  7.9 ×10-8

11. The Natural Logarithm – Base e
For all positive numbers x,
ln x = logex.
On the calculator, the natural logarithm key is usually found in conjunction with the ex key.

12. Evaluating Natural Logarithms
Example Evaluate each expression. a) ln12 b) lne10

13. Evaluating Natural Logarithms
Example Evaluate each expression. a) ln12 b) lne10 Solution a) ln12 ≈ b) lne10 = 10

14. Properties of Logarithms
Property 1 is true because a0 = 1 for any value of a. Property 2 is true since in exponential form: ak = ak. Property 3 is true since logak is the exponent to which a must be raised in order to obtain k.

15. Additional Properties of Logarithms
For x > 0, y > 0, a > 0, a  1, and any real number r,

16. Additional Properties of Logarithms
Examples Assume all variables are positive. Rewrite each expression using the properties of logarithms.

17. Using Logarithm Properties

18. Using Logarithm Properties

19. Using Logarithm Properties

20. The Change-of-Base Rule
For any positive real numbers x, a, and b, where a  1 and b  1,

21. Using the Change-of-Base Rule
Example Evaluate each expression and round to four decimal places. (a) log517 (b) log20.1

22. Using the Change-of-Base Rule
Example Evaluate each expression and round to four decimal places. (a) log517 (b) log20.1 Solution Note in the figures below that using either natural or common logarithms produce the same results.

23. 5.4 Logarithmic Functions
The logarithmic function with base a and the exponential function with base a are inverse functions. So,

24. 5.4 Graphs of Logarithmic Functions
Recall that the graph of the inverse function is reflexive about the line y = x.

25. 5.4 Graphs of Logarithmic Functions
Below are typical shapes for such graphs where 0 < a < 1.

26. Determining Domains of Logarithmic Functions
Example Find the domain of each function.
(a) f(x) = log2(x – 1) (b) f(x) = ln (x2 – 4)

27. Determining Domains of Logarithmic Functions
Example Find the domain of each function.
(a) f(x) = log2(x – 1) (b) f(x) = ln (x2 – 4)
Solution
(a) Argument of the logarithm must be positive. x – 1 > 0, or x > 1. The domain is (1, ).
(b) Use the sign graph to solve x2 – 4 > 0.

28. Graphing Translated Logarithmic Functions
Example Give the domain, range, asymptote, and x-intercept. (a) y = log2(x − 1) (b) y = (log3x) − 1

29. Graphing Translated Logarithmic Functions
Example Give the domain, range, asymptote, and x-intercept.
(a) y = log2(x − 1) (b) y = (log3x) − 1
Solution
(a) The argument x – 1 shifts the graph of y = log2 x 1 unit to the right.
Vertical asymptote: x = 1
x-intercept: (2, 0)
Domain: (1, ), Range: (–, )

30. Graphing Translated Logarithmic Functions
(b) Here, 1 is subtracted from y = log3 x shifting it down 1 unit.
Vertical asymptote: y-axis (or x = 0)
x-intercept: (3, 0)
Domain: (0, ), Range: (–, )

31. Finding the Inverse of an Exponential Function
Example Find the inverse function of f(x) = –2x + 3.

32. Finding the Inverse of an Exponential Function
Example Find the inverse function of f(x) = –2x + 3.