I want to show that $e$ (wich we will assume to be irrational) is no a root for the polynomial $p(x)=x^na-b$ with natural coeficients, in an elementary way (I'm aware that $e$ is transendental but a proof like that seems overkill for such a simple polynomial). I'm looking for an specific proof for this specific polynomial.
My attempt: By contradiction, let $e=(b/a)^{1/n}$, then $n=\log b-\log a$ (by applying $\log$ on both sides), by the fundamental theorem of calculus, $n=\int_a^b1/xdx$ and I feel like this is not possible for $a,b\in \mathbb N$ but I can't justify this intuition.
Any proofs or ideas for a proof are more than welcomed. (I don't really know what tags to use in this context, edits are welcomed).
By "specific proof" I mean any proof that proves the statement without being rediculously extensive or that relies on (considerably) higher mathematics.