Main Question or Discussion Point
i know this is possible to find with iteration, but is it possible to find it algebraically?
so if x=y^y, then y=?
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)).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).