A friend and I have been trying to prove the following:
If $x^x=y^y$, where $x,y\ge1$, prove that $x=y$.
Intuitively, it's correct, but we haven't been able to prove it. We've tried a few different ways, most of them involving logarithms, but nothing has worked as of yet. I don't think logarithms are the correct way to do this. Any help would be greatly appreciated.