A few months ago, I encountered the following problem: $$x^{x^{x^{...}}} = 2$$
The simple solution is as follows: $$x^{2} = 2 \Rightarrow x=\sqrt{2}$$.
I also discovered the following: $$x^{x^{x^{...}}} = 4 \Rightarrow x=\sqrt[4]{4} = \sqrt{2}$$
But $2 \neq 4$ and therefore this seems wrong.
On further research, I found that $\frac{1}{e} \leqslant x^{x^{x^{...}}} \leqslant e$ and thus the second case of the equation is invalid.
However, I was never really able to gauge why these bounds have been set and why one cannot set values for $x^{x^{x^{...}}} > e$.
Other than graphically, I have seen no proof for the same and would greatly appreciate any such proof or intuition that would help put things into perspective.
The problem might be in how your formula is proved. If we let $a$ be the tower of x's then after raising x to the power of the LHS and RHS we get
$x^a=x^2$. The LHS is 2 so we get $2=x^2$.
In your second equation you most likely did $2^a=2^2$, but $2 \neq x$.