If $y^y=x$, can $y$ be expressed as a function of $x$?

Question

If $y^y=x$, can y be expressed as a function of x? Specifically, I am finding the solution to a PDE where the most general solution is $u=t^{-\frac{1}{2}} f(x,t)$ and $$\LARGE f^f=Ce^{\frac{-x}{2\sqrt{t}}} $$ Any help will be appreciated!

Answer 1

If $x=y^y$, take $\log$ of both sides to get: $$y\log y=(\log y)e^{\log y} = \log x,$$ and thus $\log y = W(\log x)$ and thus $y=e^{W(\log x)}$, where $W(z)$ is the Lambert-W function satisfying $W(z)e^{W(z)}=z$.

There isn’t a way to express $W$ in terms of more usual functions, however.