ACTIVITY 37 Logarithmic Functions (Section 5.2, pp )

Definition: Let a be a positive number with a ≠ 1. The logarithmic function with base a, denoted by log a, is defined by

Properties of Logarithms (3 step proofs): Let a be a positive number with a ≠ 1

Example 1: Change each exponential expression into an equivalent expression in logarithmic form:

Example 2: Change each logarithmic expression into an equivalent expression in exponential form:

Example 3: Evaluate each of the following expressions: Property 3

Property 4 Property 3 Property 1

Graphs of Logarithmic Functions:

Example 4: Find the domain of the function and sketch its graph.

Common Logarithms: The logarithm with base 10 is called the common logarithm and is denoted by omitting the base: log x := log 10 (x).

Example 5 (Bacteria Colony): A certain strain of bacteria divides every three hours. If a colony is started with 50 bacteria, then the time t (in hours) required for the colony to grow to N bacteria is given by Find the time required for the colony to grow to a million bacteria.

Definition: Natural Logarithms The logarithm with base e is called the natural logarithm and is denoted by ln: ln x := log e x. We recall again that, by the definition of inverse functions, we have

Properties of Natural Logarithms:

Example 6: Evaluate each of the following expressions: Property 3 Property 4

Example 7: Graph the function

Example 8: Find the domain of the function