If $1<a<b$, when is $a^b>b^a$?

Question

Given $a,b\in \mathbb{N}$ where $1<a<b$, when is it so that $a^b > b^a$?

I feel like this should be a known result but I cannot for the life of me find it.

Answer 1

Over $\Bbb R$, with $1<a<b$, $a^b>b^a$ is equivalent to $b\ln a>a\ln b$, or to $(\ln a)/a>(\ln b)/b$. The function $f(x)=(\ln x)/x$ has a maximum at $e$. Thus if $e<a<b$, then certainly $a^b>b^a$. But if $1<a<e<b$, then all bets are off...