# The logarithm power rule

## Statement

The logarithm of a power is the same of the exponent multiplied by that logarithm:

$$\log_a(b^c) = c * \log_a(b)$$

## Proof

Let $x = \log_a(b)$. Now rewrite it as a power.

$$a^x = b$$

Raise both sides to the power $c$ and multiply the exponents.

$$(a^x)^c = b^c$$

$$a^{cx} = b^c$$

Take the logarithm with base $a$ on both sides. Since $\log_a$ and $a^{...}$ are inverse operations, they cancel.

$$\log_a(a^{cx}) = \log_a(b^c)$$

$$c * x = \log_a(b^c)$$

Finally, re-substitue $x$ and flip the equation.

$$c * \log_a(b) = \log_a(b^c)$$

$$\log_a(b^c) = c * \log_a(b)$$

## Proofs building upon this proof

### Exponent rule in calculus

This proofs show the derivative of a^x is a^x * ln(a).

### The difference of logarithms

This proof shows that the difference of two logarithms with the same base is just one logarithm with the inside parts being divided.