Definition

In the section on exponential functions we saw that the graph of was increasing when a > 1 and decreasing if 0 < a < 1. If you reverse the pairs on either type of graph you get the graph of the inverse function. For example in the graph below (-1, 1/2), (0, 1), (1, 2), and (2, 4) are on the graph of , which is in blue, while (1/2, -1), (1, 0), (2, 1), and (4, 2) are pairs on its inverse function pictured in red. The inverse function in this case is called the logarithm with base two.

Consider the pairs (1/2, -1), (1, 0), and (4, 2) which are on the red graph. Note that , , and that . In general (x, y) pairs on the red graph have the property that .

Definition For x >0 and a > 0 but

if and only if

This says that y is the logarithm of x with base a if and only if a to the power y equals x.

Example 1

Each of the following includes a calculation of a logarithm along with its justification.

since

since

since

since

since

since

It is important to note that