Logarithm Properties Does $$\log_{a^2}x=\log_a 2x$$
If it does, then how would you prove it?
If it doesn't, then how would I simplify $\log_{a^2} x$ so that it is in the form of $\log_a n$ where $n$ is some polynomial?

Here's an explanation which doesn't explicitly use the change of base formula:
Let $c = \log_{a^{2}}(x)$ 
Then, from the definition of the $\log$ function:
$(a^{2})^{c} = x $
$\log(a^{2c}) =\log x$
$2c\log a = \log x \Rightarrow c = \frac{\log x}{2\log a}$
Let $d = log_{a}(2x)$
Then $a^{d} = 2x$
$d = \frac{2\log x}{\log a}$
$\Rightarrow d = 4c \Rightarrow d \neq c$
