How to derive inverse of x^x to be log(x)/W(log(x))

Question

I understand the basics of the $W$ Lambert function, but I have problems working out some problems with it. I know the answer, but I don't know how to derive it. Help would be appreciated. $y^y=x$ is the same as $y=\log(x)/W(\log(x))$

Answer 1

We have \begin{align*} x &= y^y \\ \iff \log x &= y \log y \tag+\\ \iff \log x &= \log y \exp(\log y)\\ \iff W(\log x) &= \log y\\ \iff W(\log x) &= \frac{\log x}y & \text{by $(+)$}\\ \iff y &= \frac{\log x}{W(\log x)} \end{align*}