Lets ask the general question
When does $x hold for real $x,y$?
Taking logarithms, we are asking when $x implies $x\log y, or when $\frac{x}{\log x}<\frac{y}{\log y}.$ Notice this is the same as when the function $f(x)=\frac{x}{\log x}$ is increasing. To answer that we look at the derivative, and since
$f^{'}(x)=\frac{\log(x)-1}{\log^2(x)}$
we see the derivative is only positive, and that the function is only increasing, when $x>e$. Hence if $x,y>e$ you will have $x
If $0 and $x,y\neq 1$, then the derivative is actually negative, and we get the opposite $ 1x^y.$