# Solving for $x$ in $M ^ {M ^ M} = x ^ {1 / (x-1)}$ where $M = 5 ^ {\sqrt{5} / 10}$

Methods used by analogy, for example $x ^ x = 3 ^ 3 \implies x = 3$,

Determine the value of $x$ in $$M ^ {M ^ M} = x ^ {1 / (x-1)}$$ if $M = 5 ^ {\sqrt{5} / 10}$.

• Maple answers $x=2.55849460788022014674343500422$. – user64494 Aug 12 '15 at 19:03
• I think a good starting point is to recognize that $M$ can be written as $\sqrt{5}^{\frac{1}{\sqrt{5}}}$ – 1-___- Aug 12 '15 at 19:21

I'm not entirely sure how you're supposed to spot the solution here, but one may solve this equation for $x$ using the Lambert W function as follows:
$$C=M^{M^M}=x^{1/(x-1)}\\C^{x-1}=x\\\frac1C=xC^{-x}\\\frac1C=xe^{-\ln(C)x}\\-\frac{\ln(C)}C=-\ln(C)xe^{-\ln(C)x}\\W_k\left(-\frac{\ln(C)}C\right)=-\ln(C)x$$
$$x=-\frac{W_k(-\ln(C)/C)}{\ln(C)}$$
which admits two different real solutions. Since $-\ln(C)/C=\ln(1/C)e^{\ln(1/C)}$, one of these solutions simplifies down to $x=1$, which is an extraneous solution. The other is non-trivial and may simplify, though it is not obvious to me.