1. Section 3 Chapter 10

2. 1 Copyright © 2012, 2008, 2004 Pearson Education, Inc. Objectives 2 5 3 4 Logarithmic Functions Define a logarithm. Convert between exponential and logarithmic forms. Solve logarithmic equations of the form log a b = k for a, b, or k. Define and graph logarithmic functions. Use logarithmic functions in applications involving growth or decay. 10.3

3. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Define a logarithm. Objective 1 Slide 10.3- 3

4. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Logarithm For all positive numbers a, with a  1, and all positive numbers x, y = log a x means the same as x = a y. Logarithmic form: y = log a x Exponent form: x = a y Exponent Base Slide 10.3- 4 Define a logarithm.

5. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Meaning of log a x A logarithm is an exponent. The expression log a x represents the exponent to which the base a must be raised to obtain x. Slide 10.3- 5 Define a logarithm.

6. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Convert between exponential and logarithmic forms. Objective 2 Slide 10.3- 6

7. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Exponential Form Logarithmic Form Slide 10.3- 7 Convert between exponential and logarithmic forms. The table shows several pairs of equivalent forms.

8. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Fill in the blanks with the equivalent forms. Exponential Form Logarithmic Form Slide 10.3- 8 CLASSROOM EXAMPLE 1 Solution: Converting Between Exponential and Logarithmic Forms

9. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Solve logarithmic equations of the form log a b = k for a, b, or k. Objective 3 Slide 10.3- 9

10. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Solve the equation. Slide 10.3- 10 CLASSROOM EXAMPLE 2 Solving Logarithmic Equations Solution: The input of the logarithm must be a positive number. The solution set is

11. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Slide 10.3- 11 CLASSROOM EXAMPLE 2 Solving Logarithmic Equations (cont’d) The solution set is Solution: Solve the equation.

12. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Reject the negative solution since the base of a logarithm must be positive and not equal to 1. The solution set is Slide 10.3- 12 CLASSROOM EXAMPLE 2 Solving Logarithmic Equations (cont’d) Solution: Solve the equation.

13. Copyright © 2012, 2008, 2004 Pearson Education, Inc. The solution set is Slide 10.3- 13 CLASSROOM EXAMPLE 2 Solving Logarithmic Equations (cont’d) Solution: Solve the equation.

14. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Properties of Logarithms For any positive real number b, with b  1, the following are true log b b = 1 and log b 1 = 0. Slide 10.3- 14 Solve logarithmic equations of the form log a b = k for a, b, or k.

15. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Evaluate each logarithm. Slide 10.3- 15 CLASSROOM EXAMPLE 3 Using Properties of Logarithms Solution:

16. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Define and graph logarithmic functions. Objective 4 Slide 10.3- 16

17. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Logarithmic Function If a and x are positive numbers, with a  1, then g(x) = log a x defines the logarithmic function with base a. Slide 10.3- 17 Define and graph logarithmic functions.

18. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Graph f(x) = log 4 x. Write in exponential form as x = 4 y. Make a list of ordered pairs. x = 4 y y 1/16 22 1/4 11 10 41 162 Be sure to write the x and y values in the correct order. Slide 10.3- 18 CLASSROOM EXAMPLE 4 Graphing a Logarithmic Function (a > 1) Solution:

19. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Graph f(x) = log 1/10 x. Write in exponential form as Make a list of ordered pairs. x = (1/10) y y 10 11 10 1/101 1/1002 Slide 10.3- 19 CLASSROOM EXAMPLE 5 Graphing a Logarithmic Function (0 < a < 1) Solution:

20. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Characteristics of the Graph of g(x) = log a x 1. The graph contains the point (1, 0). 2. The function is one-to-one. When a > 1, the graph will rise from left to right, from the fourth quadrant to the first. When 0 < a < 1, the graph will fall from left to right, from the first quadrant to the fourth quadrant. 3. The graph will approach the y-axis, but never touch it. (The y-axis is an asymptote.) 4. The domain is (0,  ), and the range is ( ,  ). Slide 10.3- 20 Define and graph logarithmic functions.

21. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Use logarithmic functions in applications involving growth or decay. Objective 5 Slide 10.3- 21

22. Copyright © 2012, 2008, 2004 Pearson Education, Inc. A population of mites in a laboratory is growing according to the function defined by P(t) = 80 log 10 (t + 10), where t is the number of days after a study is begun. Find the number of mites at the beginning of the study. Find the number present after 90 days. Slide 10.3- 22 CLASSROOM EXAMPLE 6 Solving an Application of Logarithmic Function

23. Copyright © 2012, 2008, 2004 Pearson Education, Inc. P(t) = 80 log 10 (t + 10) Find the number of mites at the beginning of the study. The number of mites at the beginning of the study is 80. Slide 10.3- 23 CLASSROOM EXAMPLE 6 Solving an Application of Logarithmic Function (cont’d) Solution:

24. Copyright © 2012, 2008, 2004 Pearson Education, Inc. P(t) = 80 log 10 (t + 10) Find the number present after 90 days. The population after 90 days is 160 mites. Slide 10.3- 24 CLASSROOM EXAMPLE 6 Solving an Application of Logarithmic Function (cont’d) Solution: