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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.

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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*}

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  • $\begingroup$ I think a proof (which is not hard at all) would make this a great answer. $\endgroup$ – Don Thousand Oct 17 '19 at 1:16
  • $\begingroup$ @DonThousand I'll edit my answer $\endgroup$ – Luke Collins Oct 17 '19 at 1:25

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