How to prove that $\frac{\ln 5}{\ln 2}$ is irrational

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I have to show that $\frac{\ln 5}{\ln 2}$ is irrational

I have tried the following:

Assume it's rational so

$\frac{\ln 5}{\ln 2} = \frac{p}{q}$

which becomes

$\log _2\left(5\right)=\frac{p}{q}$

Therefore

$5=2^{\frac{p}{q}}$

This is where I'm stuck, I don't know how to advance my proof any further

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1 Answer

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$$\frac{\log 5}{\log 2}=\frac pq\iff 5^q=2^p$$ for naturals $p,q$. By the fundamental theorem of arithmetic and primality of $2$ and $5$, this is not possible.

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