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