0
$\begingroup$

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 $M = 5 ^ {\sqrt{5} / 10}$.

$\endgroup$
  • 1
    $\begingroup$ Maple answers $x=2.55849460788022014674343500422 $. $\endgroup$ – user64494 Aug 12 '15 at 19:03
  • $\begingroup$ I think a good starting point is to recognize that $M$ can be written as $\sqrt{5}^{\frac{1}{\sqrt{5}}}$ $\endgroup$ – 1-___- Aug 12 '15 at 19:21
2
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ Do you think the question is a PSQ? If you do then try not to give a full answer, even though I see that you've been quite into the Lambert W function lately :) $\endgroup$ – TheSimpliFire Aug 12 '18 at 16:08
  • $\begingroup$ @TheSimpliFire I do, though it's old and I'm trying to remove it from the unanswered list. (One may also wish to see the OP's profile) $\endgroup$ – Simply Beautiful Art Aug 12 '18 at 16:10
  • $\begingroup$ Mhm... In any case do as you wish with them. $\endgroup$ – Simply Beautiful Art Aug 12 '18 at 16:12

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.