# Logarithms

A logarithm of a number to a given base is defined as the power to which the base is raised in order to produce that number. In simpler terms, a logarithm is the opposite of an exponent: A logarithm is the operation you perform to undo an exponent. Just as division on a number would undo whatever multiplication was done to the original number, logarithms and exponents have a similar relationship.

A logarithm is expressed as follows:

where **a** is known as the base, **x** and **y** are defined by the equation
below:

from which you can see that **y** is the power to which the base **a** is
raised, in order to get the number whose logarithm we were trying to find. Note
that **x** MUST be positive (i.e. non-negative) and NOT zero!

To prove the above equation, first consider the following simple logarithmic identities

In the above identities where the base is not explicitly shown, consider the base to be 10.

If we were to start with the following equation:

then taking the logarithm of the above expression to some arbitrary base

this can also be expressed as

and from this we can see that

## Logarithm Examples

Example 1: Evaluate the following

*solution*

Example 2: Find **a** in the following equation

*solution*

## Logarithmic Identities

See Logarithmic Functions in Pre-Calculus for help with functions involving logarithms.