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

[3hlang]

i know this is possible to find with iteration, but is it possible to find it algebraically?

 

[tiny-tim]

Hi 3hlang!

i know this is possible to find with iteration, but is it possible to find it algebraically?

Nope!

 

[HallsofIvy]

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).

 

[g_edgar]

... and the reason we talk about the Lambert W function is that none of the previously-defined standard functions would work.

 

[3hlang]

so if x=y^y, then y=?

 

[HallsofIvy]

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 x= y^y then, by the same formula I gave before, y= W(ln(x)).

 

[3hlang]

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