# What is the exponent rule of logarithms?

Sep 16, 2016

${\log}_{a} \left({m}^{n}\right) = n {\log}_{a} \left(m\right)$

#### Explanation:

Consider the logarithmic number ${\log}_{a} \left(m\right) = x$:

${\log}_{a} \left(m\right) = x$

Using the laws of logarithms:

$\implies m = {a}^{x}$

Let's raise both sides of the equation to $n$th power:

$\implies {m}^{n} = {\left({a}^{x}\right)}^{n}$

Using the laws of exponents:

$\implies {m}^{n} = {a}^{x n}$

Let's separate $x n$ from $a$:

$\implies {\log}_{a} \left({m}^{n}\right) = x n$

Now, we know that ${\log}_{a} \left(m\right) = x$.

Let's substitute this in for $x$:

$\implies {\log}_{a} \left({m}^{n}\right) = n {\log}_{a} \left(m\right)$