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

$$2^{log_2 3} \Leftrightarrow 2^{\frac{3}{2}}$$

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

$$3^2 \Leftrightarrow 2^3$$

and see that

$$9>8$$

therefore

$$log_2 3>\frac{3}{2}$$

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

$$3^{\frac{3}{2}} \Leftrightarrow 3^{log_3 5}$$

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

$$3^3 \Leftrightarrow 5^2$$ $$27 > 25$$

therefore

$$\frac{3}{2} > log_3 5$$

Finally

$$log_2 3 > \frac{3}{2} > log_3 5$$

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