How do you explicitly represent $x^{x^y}=y^{y^x}$ using the Lambert $W$ function?
I started using logarithms to split it up and manipulate it to a form like xe^x. I do this semi-successfully. I go through the steps and get $e^{xln(y)} *ln(y) = e^{yln(x)} *ln(x)$ and basically get stuck. I can multiply $x$ and $y$ to both sides but I still have a problem with having both variables on each side. I may be going about this incorrectly for this type of problem.
If we look at the definition of the Lambert W function, we see that the Lambert W function is the inverse of something that goes up to what I will call the second floor of a power tower.
$$xe^x\qquad \text{$\leftarrow$ the $x$ in the exponent is on the second floor.}$$
Now, if I wanted to solve something where a variable was in the first, second, and third floor, I would have to use something beyond the Lambert W function, as the Lambert W function can only solve things (generally, there are some special cases) where the variable is in $2$ consecutive floors. Special cases arrive when we have something like $xe^xe^{xe^x}$, where we see it has the form $f(x)e^{f(x)}$, but if this is not so obvious, it is probably not solvable.
So I'd have to say that it is not possible to solve for $x$ or $y$ here.
If the solution is findable in terms of the Lambert W function, generally make the substitution $x=W(u)$ and use $e^{W(u)}=\frac u{W(u)}$ to reduce the amount of "floors" there are.