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

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

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  • $\begingroup$ By the way, the $x$ that solves $2^x=3$ has a special symbol: it is called $\log_23$. That is, $2^{\log_23}=3$. $\endgroup$ – Akiva Weinberger Oct 16 '15 at 13:14
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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 $2^p$.

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