There are plenty of discussion on MSE about how to compare $x^y$ and $y^x$. For $x,y>e$, it is sufficient to just compare $x$ and $y$ to reach a conclusion. But I wonder if there are some general steps that can be used in any situations where $x<e$ and $y>e$. I tried to use logarithm, but the inequities produced often includes multiplication of negative numbers, so I often make mistakes along the way.
Note that I am interested in all REAL values of $x,y$, NOT just integers. For example, $x=2,y=\sqrt{5}$. I wish anyone to give a solution that is applicable in other values of $x,y$ as well.