Indices confusion

Question

Consider the equation $3^{y} = 9^{x}$

It follows that $3^{y} = 3^{2x}$

But $3^{2x} \equiv (3^{x})^{2} \equiv (3^{2})^{x}$ (I think? Since e.g. $(x^{2})^{3} \equiv x^{2 \cdot 3} \equiv (x^{3})^{2}$ right?)

So which of the following is correct? $y = 2x$ or $y = x^2$ or $y = 2^x$?

Thanks!

Answer 1

$y=2x$ is correct. $(3^2)^x=3^y$ is also correct but does not imply $y=2^x$ or something similar.

The confusion might arise from the fact that powering is not associative: i.e. in general it is not true that $a^{(b^c)}=(a^b)^c$, e.g. $3^{(2^3)}\ne (3^2)^3$.

Answer 2

$y=2x$ is the correct option. Because if $$x^{a}=x^{b} \Longrightarrow a=b$$