A question of rationality of integral powers

Question

Given $a,b\in\mathbb{Z}$, is $x$ from $a^x=b$ ever rational?

More specifically, is $x$ in $2^x=3$ irrational?

Answer 1

For the first question, $x$ can certainly be rational, for $2^2=4$ and $8^{1/3}=2$.

For $2^x=3$, if $x=p/q$ where $p$ and $q$ are integers, with say $q$ positive, we get $2^p=3^q$, which is impossible since $3$ does not divide $2^p$.