# Logarithm Function

## Function

`log(x,b)`

## Summary

Returns the logarithm with a base of b of the number x.

## Input

Name | Description |
---|---|

x | the input number |

b | the base of the logarithm |

## Output

` log(8,2) = 3`

` log(9,3) = 2`

` log(27,3) = 3`

` log(1,4) = 0`

` log(4,4) = 1`

` log(16,4) = 2`

` log(81,9) = 2`

## Explanation

The logarithm function is the inverse operation of exponentiation. This is expressed by the two equations below, in both cases the variable represents the base of the operation.

For example, 2 raised to the 3rd power is equal to 8. Inversely, the logarithm base 2 of 8 is equal to 3.

### Notation

Sometimes the base of the logarithm is written as subscript after the function name.

### Graph of Logarithm Base 2

Below is the graph of the logarithm function with a base of 2. The logarithm with any base, , of 1 will always equal 0. Note, for everytime the input increases by a factor of 2 the output of the function increases by 1.

` log(1,2) = 0`

` log(2,2) = 1`

` log(4,2) = 2`

` log(8,2) = 3`

### Graph of Logarithm Base 3

Below is the graph of the logarithm function with a base of 3. Note, everytime the input increases by a factor of 3 the output of the function increases by 1.

` log(1,3) = 0`

` log(3,3) = 1`

` log(9,3) = 2`

` log(27,3) = 3`

## Natural Logarithm Function

The natural logarithm function is defined as the logarithm function with a base (Euler’s Number).

Sometimes the notation refers to the natural logarithm, but it can also refer implicitly to the . This site will always use the notation or to be explicit.