1. Precalculus Essentials
Fifth Edition
Chapter 3
Exponential and Logarithmic Functions
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2. 3.2 Logarithmic Functions

3. Objectives 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.

4. Definition of the Logarithmic Function
For x > 0 and b > 0, b ≠ 1,

5. Example: Changing from Logarithmic to Exponential Form
Write each equation in its equivalent exponential form:

6. Example: Changing from Exponential to Logarithmic Form
Write each equation in its equivalent logarithmic form:

7. Example 1: Evaluating Logarithms
Solution:

8. Example 2: Evaluating Logarithms
Solution:

9. Example 3: Evaluating Logarithms
Solution:

10. Basic Logarithmic Properties Involving One

11. Example: Using Properties of Logarithms
Solution:

12. Inverse Properties of Logarithms
For b > 0 and b ≠ 1,

13. Example: Using Inverse Properties of Logarithms
Solution:

14. Example: Graphs of Exponential and Logarithmic Functions (1 of 3)

15. Example: Graphs of Exponential and Logarithmic Functions (2 of 3)

16. Example: Graphs of Exponential and Logarithmic Functions (3 of 3)

17. Characteristics of Logarithmic Functions of the Form f(x) = logb x

18. The Domain of a Logarithmic Function

19. Example: Finding the Domain of a Logarithmic Function
Solution:

20. Example: Illustration of the Domain of a Logarithmic Function
Solution: We found that the domain of f is (5, ∞). This is illustrated by the graph of f.

21. Common Logarithms
The logarithmic function with base 10 is called the common logarithmic function.

22. Example 1: Application
The percentage of adult height attained by a boy who is x years old can be modeled by f(x) = log(x + 1), 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. Properties of Common Logarithms

24. Natural Logarithms
The logarithmic function with base e is called the natural logarithmic function.

25. Properties of Natural Logarithms

26. Example 2: 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 f(x) = 13.4 ln x − 11.6 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.