8.4 Logarithms and Logarithmic Functions

8.4 Logarithms and Logarithmic FunctionsGoal: Evaluate and graph logarithmic functionsCorrect Section 8.3Warm-upFind the inverse: y = 3x - 5If 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.

Logarithms

because

Definition of Logarithm with Base bDefinition of Logarithm with Base bLet 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.Example 1Rewrite the equation in exponential form:

Special Logarithm ValuesSpecial Logarithm ValuesLet b be a positive number such that b 1.Logarithm of 1

Logarithm of Base b

Example 2Evaluate 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 logarithmsA common logarithm is a logarithm with base 10. A natural logarithm is a logarithm with base e.

Evaluate:

Inverse functionsRemember 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

Example 3Evaluate the expressions:

Example 4Evaluate the expressions:

Graph

Graph

Domain: x > 0Range: All Real NumbersGraph

Domain: x > 0Range: All Real NumbersGraph

Domain: x > 1Range: All Real Numbers

AssignmentWorksheet 8.4How Do I LiveLeAnn Rimes

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