1. 8.4 Logarithms and Logarithmic Functions Goal: Evaluate and graph logarithmic functions Correct Section 8.3

2. Warm-up Find the inverse: y = 3x - 5 If f(x) = 2x and g(x) = 3x – 4, find f(g(3)) An account that pays 3% annual interest compounded continuously has a balance of $10,000 on June 1, 2008. If no money is added, what is the balance on June 1, 2010? There will be about $10,618.37 in the account.

3. Logarithms because

4. Definition of Logarithm with Base b Let b and y be positive numbers such that b ≠ 1. The logarithm of y with base b is denoted by and is defined as follows: If b is not given, it is 10. This is known as a common logarithm.

5. Example 1 Rewrite the equation in exponential form:

6. Special Logarithm Values Let b be a positive number such that b ≠ 1. Logarithm of 1 Logarithm of Base b

7. Example 2 Evaluate the expression: Rewrite the expression in exponential form: Rewrite the right hand side so it is the same base as the left hand side of the equation:

8. Evaluate the expression:

9. Special logarithms A common logarithm is a logarithm with base 10. A natural logarithm is a logarithm with base e. Evaluate:

10. Inverse functions Remember that the functions f and g are inverses of each other if f(g(x)) = x and g(f(x)) = x The logarithmic function is the inverse of the exponential function Therefore: and

11. Example 3 Evaluate the expressions:

12. Example 4 Evaluate the expressions:

13. Graph

14. Domain: x > 0 Range: All Real Numbers

15. Graph Domain: x > 0 Range: All Real Numbers

16. Graph Domain: x > 1 Range: All Real Numbers

17. Assignment Worksheet 8.4