TITLE: Seems that I just proved $2=4$.
QUESTION [27 upvotes]: Solving $x^{x^{x^{.^{.^.}}}}=2\Rightarrow x^2=2\Rightarrow x=\sqrt 2$.
Solving $x^{x^{x^{.^{.^.}}}}=4\Rightarrow x^4=4\Rightarrow x=\sqrt 2$.
Therefore, $\sqrt 2^{\sqrt 2^{\sqrt 2^{.^{.^.}}}}=2$ and $\sqrt 2^{\sqrt 2^{\sqrt 2^{.^{.^.}}}}=4\Rightarrow\bf{2=4}$.
What's happening!?

REPLY [2 votes]: In general, you can solve the equation in terms of the Lambert W function as
$$ y=x^{x^{x^{.^{.^.}}}} \implies \ln(y)=y\ln(x) \implies y = -\frac{W(-\ln(x))}{\ln(x)}.  $$
Try to use this closed form to see what the problem is. Note this, if you ask maple to solve the equations
$$  -\frac{W(-\ln(x))}{\ln(x)}=2,\quad  -\frac{W(-\ln(x))}{\ln(x)}=4, $$
you will get the same answers
$$ x=1,\sqrt{2} $$
More generally, the solution of 
$$ -\frac{W(-\ln(x))}{\ln(x)}=a $$
is given by
$$ \left\{ x=1,x={{\rm e}^{{\frac {\ln  \left( a \right) }{a}}}}\right\}. $$
Now, if you use the above solution with $a=2,4$, you will see why?