## Logarithmic Functions

A logarithmic function is the inverse of the exponential function. In maths methods we mostly focus on the natural log which is the inverse of the euler function:

$log_{e}(x)$

However, sometimes we do use log base 10 which means for every increase of 1 in x the value of the log increases by a factor of 10.

## Logarithmic differentiation

The logarithmic differentiation formula can be found in the formula sheet:

$\frac{d}{dx}(log_{e}(x))=\frac{1}{x}$

This is one of the more tricky differentiation processes that we have to do. So to simplify it we can split it up into two processes, calculating the numerator and calculating the denominator.

### Denomintor

The denominator is the bottom part of the fraction. It is relatively easy to get since all you are doing is taking whatever is inside the logs brackets and that becomes the denominator.

### Numerator

The numerator is the top part of the fraction. It is calculated by calculating the derivative of whatever is in the brackets. Sometimes this will just be power functions which we have already learnt how to calculate but other time we will need to use more complicated rules such as the product rule, quotient rule and chain rule.

Lets take a look at an actual example

### Logarithmic Differentiation

Find the derivative of:

$log_{e}(2x+5)$

Our first step is to find the denominator which we get by taking what is inside the brackets.

$\frac{numerator}{2x+5}$

The second step is to find the numerator, by finding the derivative of the terms within the bracket.

$numerator= \frac{d}{dx}(2x+5)\newline =2$

Now all we need to do is place the numerator on top of the denominator calculated earlier.

$\frac{dy}{dx} =\frac{2}{2x+5}$