# Inverse of x^x? [duplicate]

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.

## marked as duplicate by Jack, Community♦Oct 17 at 3:29

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*}