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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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5  
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

2 Answers 2

up vote 16 down vote accepted

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

You should try WolframAlpha for similar problems. WolframAlpha would solve y=x^x for y=5 as shown here (using Lamber W Function as suggested before).

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3  
If you're just going to post a link to wolframalpha, you could at least make sure that it works... –  t.b. Dec 31 '11 at 16:20
1  
If a natural number y is entered in "solve x^x=y" WA will give the numeric answer. Strangely it doesn't work for all real y. However, I downvoted since the link is no help in understanding how the solution was reached. –  David Marquis Dec 31 '11 at 17:22
2  
I guess part of GSBabil's point is that Josh should have tried that first, before posting here. –  GEdgar Dec 31 '11 at 18:33

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