Inverse of x^x?

Question

What is the inverse of $y = x^x$? I know it won't be in terms of elementary functions, but I believe some function analogous to Lambert W function for $z = ye^y$ would be defined? I couldn't find it on google.

Answer 1

You can use the Lambert-$W$: $$y^{-1}(x)=e^{W(\log x)} = \frac{\log x}{W( \log x)}$$

Proof: \begin{align*} y=x^x &\iff \log y=x\log x\\ &\iff \log y=e^{\log x}\log x\\ &\iff W(\log y)=\log x\\ &\iff x=e^{W(\log y)}. \end{align*}