## What is the logarithm

The logarithm is the opposite of exponentiation. If we consider an equation , then in the case of exponentiation we wanted to know the value of , given the values of and . In the case of logarithms, we are given and and we want to find the exponent that solves the equation: for which value will ?

The logarithm is a function that answers this question and gives us the value for :

would be read as “the log base 5 of “.

### Examples of logarithms

(1)

## Natural logarithm

We write the logarithm to a base of a number as . When we don’t write the base and only write then we assume that the base is 10. There is one special logarithm called the natural logarithm with base (Euler’s number ). The natural logarithm of a number can be written as before, but more common is to write it as .

## Using your calculator for logarithms of any base

Your calculator will have two buttons, one for (base ) and one for (base 10). If you want to solve, for example , you can use either the logarithm with base 10 or with base but you need to follow a rule.

For base 10:

So if we remember that your button on your calculator assumes a base of 10, then to solve we solve:

For base : The same rule applies as above and we can solve like this: