Proving $x=2,y=4$ is the only solution to $x^y=y^x$ Prove $x=2,y=4$ is the only solution to $x^y=y^x$ with the additional proviso that $x\ne y$ and $x,y$ are positive integers (if $(x,y)$ is a solution, so is $(-x,-y)$). Ideally I am looking for a very simple proof - we can exhaust the possibilities below, and other possibilities look remote.

Take logarithms. You want to analyze $f(t)=t^{-1}\log t$ and its repeated values. If you consider the derivative $f'(t)$ you should conclude that $f$ increases from $0$ to $t=e$ and then decreases to $0$ at $t=+\infty$. Try to play around with this a bit and see what it gives you.