# Change of base formula for logarithms

## Statement

Every logarithm can be written as a division of two logarithms with a custom base:

$$\log_a(b) = \frac{\log_c(b)}{\log_c(a)}$$

## Proof

Let $x$, $y$ and $z$ be logartihms and write them as exponentials.

$$x = \log_a(b) \implies b = a^x$$

$$y = \log_c(b) \implies b = c^y$$

$$z = \log_c(a) \implies a = c^z$$

From $b = a^x$ and $b = c^y$ follows the following.

$$a^x = c^y$$

Now substitute $a = c^z$.

$$(c^z)^x = c^y$$

Multiply the exponents and set them equal to eachother.

$$c^{xz} = c^y$$

$$xz = y$$

Divide both sides by $z$ to isolate $x$.

$$x = \frac{y}{z}$$

Finally, re-substitue $x$, $y$ and $z$.

$$\log_a(b) = \frac{\log_c(b)}{\log_c(a)}$$

## Proofs building upon this proof

### Log rule in calculus

This proofs shows the derivative of a logarithmic function.