## Power law

`log`

_{b}p^{n} = n * log_{b}p

**Proof:**

`log`

-> _{b}p = a`b`

^{a} = p

Raise to `k`

-> `b`

^{k*a} = p^{k}

Apply `log`

-> _{b}`k * a = log`

_{b}p^{k}

Replace `a`

with `log`

-> _{b}p`k * log`

_{b}p = log_{b}p^{k}

## Product law

`log`

_{b}m*n = log_{b}m + log_{b}n

**Proof:**

`log`

-> _{b}m = i`b`

& ^{i} = m`log`

-> _{b}n = j`b`

^{j} = n

Multiply -> `b`

^{i} * b^{j} = m * n

Simplify -> `b`

^{i+j} = m * n

Apply `log`

-> _{b}`i + j = log`

_{b}m*n

Replace `i`

and `j`

-> `log`

_{b}m + log_{b}n = log_{b}m*n

## Quotient law

`log`

_{b}m/n = log_{b}m - log_{b}n

**Proof:**

`log`

-> _{b}m = i`b`

& ^{i} = m`log`

-> _{b}n = j`b`

^{j} = n

Divide -> `b`

^{i} / b^{j} = m / n

Simplify -> `b`

^{i-j} = m / n

Apply `log`

-> _{b}`i - j = log`

_{b}m/n

Replace `i`

and `j`

-> `log`

_{b}m - log_{b}n = log_{b}m/n