# Log Calculator (Logarithm)

This log calculator (logarithm calculator) allows you to calculate a logarithm of any number with any arbitrary base. Regardless of whether you are looking for a natural logarithm, log base 2, or log base 10, this tool will solve your problem for you. Read on to get a better understanding of the logarithm formula and the rules you have to follow.

## What is a logarithm?

A logarithmic function is an inverse of the exponent function. In essence, if **a** raised to power **y** gives **x**, then the logarithm of **x** with base **y** is equal to **x**. In the form of equations,

`a^y = x`

is equivalent to `log_a(x) = y`

If you want to calculate the natural logarithm of any number, you need to choose a base equal to the number e = 2.718. The natural logarithm is denoted with symbol ln(x).

One of the popular bases for logarithms is 10. The logarithm with base 10 is denoted as lg(x). It is used, for example, in our decibel calculator.

## How to calculate logarithm with arbitrary base

If you want to calculate a logarithm with an arbitrary base, but are able to access only a natural logarithm calculator or a log 10 base calculator, you need to apply the following rules:

`log_a(x) = ln(x) / ln(a)`

`log_a(x) = lg(x) / lg(a)`

We listed some basic log rules for operations on logarithms below as well.

`log_a(x*y) = log_a(x) + log_a(y)`

`log_a(x/y) = log_a(x) - log_a(y)`

`log_a(x^y) = y*log_a(x)`

## Log base 2: an example

Let's assume you want to use this tool as a log base 2 calculator. To calculate the logarithm of any number, simply follow these steps:

- Decide on the number you want to find the logarithm of. Let's say it's 100.
- Decide on your base - in this case, 2.
- Find the logarithm with base 10 of number 100.
`lg(100) = 2`

. - Find the logarithm with base 10 of number 2.
`lg(2) = 0.30103`

. - Divide these values by one another:
`lg(100)/lg(2) = 2 / 0.30103 = 6.644`

. - You can also skip steps 3-5 and input the number and base directly into the log calculator.