# Is $x^x=y$ solvable for $x$?

Given that

1. $x^x = y$; and
2. given some value for $y$

is there a way to expressly solve that equation for $x$?

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Take logs, set $x = e^t$ and apply this: math.stackexchange.com/questions/10261/inverse-of-y-xex –  Aryabhata Jul 28 '11 at 7:03

As Aryabhata mentions this is another application for the Lambert W function. The solution to your problem is presented in the wikipedia article. Using elementary substitutions you have

$$x=\frac{\ln(y)}{W(\ln y)}$$

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Wow - thanks - I really didn't think it was possible. –  Josh Jul 28 '11 at 22:23