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
Example 1: Evaluate the following
Example 2: Find a in the following equation