8 4 Logarithms and Logarithmic Functions Goal Evaluate

8. 4 Logarithms and Logarithmic Functions Goal: Evaluate and graph logarithmic functions Correct Section 8. 3

Warm-up Find the inverse: y = 3 x - 5 If f(x) = 2 x and g(x) = 3 x – 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.

Logarithms because

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 defined as follows: and is If b is not given, it is 10. This is known as a common logarithm.

Example 1 Rewrite the equation in exponential form:

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

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:

Evaluate the expression:

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

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 the exponential function Therefore: and is the inverse of

Example 3 Evaluate the expressions:

Example 4 Evaluate the expressions:

Graph

Graph Domain: x > 0 Range: All Real Numbers

Graph Domain: x > 0 Range: All Real Numbers

Graph Domain: x > 1 Range: All Real Numbers

Assignment Worksheet 8. 4