1. Chapter 4 Exponential and Logarithmic Functions Copyright © 2014, 2010, 2007 Pearson Education, Inc. 1 4.2 Logarithmic Functions

2. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 2 Change from logarithmic to exponential form. Change from exponential to logarithmic form. Evaluate logarithms. Use basic logarithmic properties. Graph logarithmic functions. Find the domain of a logarithmic function. Use common logarithms. Use natural logarithms. Objectives:

3. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 3 Definition of the Logarithmic Function For x > 0 and b > 0, b 1, y = log b x is equivalent to b y = x. The function f(x) = log b x is the logarithmic function with base b.

4. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 4 Example: Changing from Logarithmic to Exponential Form Write each equation in its equivalent exponential form: means

5. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 5 Example: Changing from Exponential to Logarithmic Form Write each equation in its equivalent logarithmic form: means

6. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 6 Example: Evaluating Logarithms Evaluate: means 10 to what power gives 100? because 10 2 = 100 Evaluate: means 5 to what power gives because

7. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 7 Example: Evaluating Logarithms Evaluate: means 36 to what power gives 6? because

8. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 8 Example: Evaluating Logarithms Evaluate: means 3 to what power gives because

9. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 9 Basic Logarithmic Properties Involving One 1. log b b = 1 because 1 is the exponent to which b must be raised to obtain b. (b 1 = b) 2. log b 1 = 0 because 0 is the exponent to which b must be raised to obtain 1. (b 0 = 1)

10. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 10 Example: Using Properties of Logarithms Evaluate: log 9 9 Because log b b = 1, we conclude log 9 9 = 1. Evaluate: log 8 1 Because log b 1 = 0, we conclude log 8 1 = 0.

11. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 11 Inverse Properties of Logarithms For b > 0 and b 1,

12. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 12 Example: Using Inverse Properties of Logarithms Evaluate: Because we conclude Evaluate: Because we conclude

13. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 13 Example: Graphs of Exponential and Logarithmic Functions Graph and in the same rectangular coordinate system. x – 2 –1–1 0 1 We first set up a table of coordinates for We will reverse these coordinates for the inverse function

14. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 14 Example: Graphs of Exponential and Logarithmic Functions (continued) We are graphing and x – 2 –1–1 0 1 x –1 10 01

15. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 15 Example: Graphs of Exponential and Logarithmic Functions (continued) We now graph and in the same rectangular coordinate system.

16. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 16 Characteristics of Exponential Functions of the Form

17. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 17 Characteristics of Logarithmic functions of the Form

18. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 18 The Domain of a Logarithmic Function The domain of an exponential function of the form includes all real numbers and its range is the set of positive real numbers. Because the logarithmic function reverses the domain and the range of the exponential function, the domain of a logarithmic function of the form is the set of all positive real numbers. In general, the domain of consists of all x for which g(x) > 0.

19. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 19 Example: Finding the Domain of a Logarithmic Function Find the domain of The domain of f consists of all x for which x – 5 > 0. Thus, the domain of f is

20. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 20 Example: Finding the Domain of a Logarithmic Function (continued) Find the domain of We found that the domain of f is This is illustrated by the graph of f. Vertical asymptote x = 5 All points on the graph of f have x-coordinates that are greater than 5.

21. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 21 Common Logarithms The logarithmic function with base 10 is called the common logarithmic function. The function is usually expressed

22. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 22 Example: Application The percentage of adult height attained by a boy who is x years old can be modeled by where x represents the boy’s age (from 5 to 15) and f(x) represents the percentage of his adult height. Approximately what percentage of his adult height has a boy attained at age ten? A ten-year old boy has attained approximately 80% of his adult height.

23. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 23 Properties of Common Logarithms

24. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 24 Natural Logarithms The logarithmic function with base e is called the natural logarithmic function. The function is usually expressed

25. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 25 Properties of Natural Logarithms

26. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 26 Example: Application When the outside air temperature is anywhere from 72° to 96° Fahrenheit, the temperature in an enclosed vehicle climbs by 43° in the first hour. The function models the temperature increase, f(x), in degrees Fahrenheit, after x minutes. Use the function to find the temperature increase, to the nearest degree, after 30 minutes. The temperature will increase by approximately 34° after 30 minutes.