intuition for basic inequality involving exponentials

Question

If $m<n$, how do the two quantities compare: $m^n$ and $n^m$?

Example: $2^5>5^2$, $3^7>7^3$, ... and so on. Is it true in general as in is there a sound proof?

Answer 1

As you've written it, it isn't true in general. You can take $m=2<4=n$ as a counterexample since $2^4=4^2$. However, if $3\leq m<n$, then we have $m^n>n^m$.

To see this, take the logarithm of both sides and reduce, so we have $$\frac{\log m}{m}>\frac{\log n}{n}.$$This inequality is true since $f(x)=\frac{\log x}{x}$ has a negative derivative for $x>e$ (which implies $f(x)$ is decreasing).