If y=x^x then what is x in terms of y?

  • Thread starter 3hlang
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  • #1
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i know this is possible to find with iteration, but is it possible to find it algebraically?
 

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  • #2
tiny-tim
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Hi 3hlang! :smile:
i know this is possible to find with iteration, but is it possible to find it algebraically?
Nope!
 
  • #3
HallsofIvy
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This probably isn't what you mean by "algebraically" but if you take the logarithm of both sides of the equation you get ln(y)= xln(x) and then x= W(ln(y)) where W is the "Lambert W function" which is defined as the inverse function to xln(x).
 
  • #4
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... and the reason we talk about the Lambert W function is that none of the previously-defined standard functions would work.
 
  • #5
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so if x=y^y, then y=?
 
  • #6
HallsofIvy
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so if x=y^y, then y=?
This probably isn't what you mean by "algebraically" but if you take the logarithm of both sides of the equation you get ln(y)= xln(x) and then x= W(ln(y)) where W is the "Lambert W function" which is defined as the inverse function to xln(x).
All you have done is swap x and y! If [itex]x= y^y[/itex] then, by the same formula I gave before, y= W(ln(x)).
 
  • #7
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what? i wasn't claiming to have done anything radical to the formula. all i wanted to know was what you do the w function of. but thank you for telling me
 

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