What is the logarithm

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

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


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

Examples of logarithms

(1)   equation

Natural logarithm

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

Using your calculator for logarithms of any base

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

For base 10:


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


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