Can this be proved that $\log(n)$ is irrational for every $n=1,2,3,\dots$ ?
I find that question in my mind in searching for if $\log(x)$ is irrational for every rational number $x\gt0$.
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2This follows from the fact that $e$ is transcendental – MPW Aug 24 '15 at 14:15
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Related. – Caleb Stanford Aug 24 '15 at 15:50
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Let $n\in\mathbb{N}$, $n\neq 1$. Suppose that there exist $a,b\in\mathbb{Z}$ such that $\log(n)=\frac{a}{b}$. Then we have that $n=e^{a/b}$ is a trancendental number since $e$ is a trancendental number and $\frac{a}{b}$ is rational. This is a contradiction. Therefore, $\log(n)$ is irrational for every $n=2,3,4,....$
Ben Sheller
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Excellent! You can in fact take $n\in\mathbb Q$ without changing anything to completly answer OP's question. – Scientifica Aug 24 '15 at 14:57
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There is a theorem (mentioned here) that for any nonzero $\alpha$, at least one of $\alpha$ and $e^{\alpha}$ is transcendental.
In this case, take $\alpha = \log x$, where $x$ is rational. Then either $\log x$ is transcendental, or $x$ is. But $x$ is rational, hence algebraic. So $\log x$ is transcendental, and in particular it is irrational.
Caleb Stanford
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