Summary

Logarithmic functions ``undo'' what exponential functions do: in fact they're defined this way.

Definition: Logarithm of base a: For a>0, , and x>0,

The logarithmic function of base a is , for x>0.

If we look at the plot of the logarithmic function of base a corresponding to a given exponential function of base a

we notice that the graphs are mirror images of each other about the line y=x. Hence, if you understand the exponential function well, you have a really good grasp of the corresponding logarithmic function.

As was the case for exponential functions, a log function is one-to-one: if x>0, y>0, b>0, and , then

More generally, all the properties of exponential functions are ``mirrored'' in the properties of the logarithmic functions. Let x and y be any positive real numbers, and r any real number. Let a be the base of an exponential function.

Popular bases for logarithms are 2, 10, and especially e. The logarithmic function of base e is so important that it is given a special name: the natural logarithm, designated . Most calculators have a special button for calculating natural logs.

It's easy to change from one base to another base for logarithms:

In particular, using the natural log ,

Using the natural logarithm, we can also generate a formula for the change of base theorem for exponential functions: