What is greater, $log_2 3$ or $log_3 5$?

The trick here is to compare both to $\frac{3}{2}$ rather than to each other. Start by comparing $log_2 3$ to $\frac{3}{2}$. Function $f(x)=2^x$ is monotonic so we can apply it to both sides to get

Function $f(x)=x^2$ is also monotonic so we can apply it to both sides

and see that

therefore

Now compare $\frac{3}{2}$ to $log_3 5$. Function $f(x)=3^x$ is monotonic so we can apply it to both sides

Function $f(x)=x^2$ is also monotonic so we can apply it to both sides

therefore

Finally

Source: problem 17 in https://arxiv.org/pdf/1110.1556v2.pdf