**Edit:**

My question has been requested to close due to its apparent lack of clarity. My question is below under "**Problem**". If the information above it is redundant, please let me know in a comment. I will try my best to make my question as clear as possible.

I was looking at the equation $x^y=y$. If I want to isolate $x$, we would have $$x=y^{1/y}\stackrel{\small\rm{or}}{=}\sqrt[y]{y}.$$

Now let's consider $x^{x^y}=y$. I did not know how to do this, but I noticed something at first. If we let $x^y=y$, then this means $x^{x^y}=y\implies x^y=y$ by order of substitution. Notice that the same equation we substituted is in fact yielded; and, as mentioned previously, this equation would arrive at the fact that $x=y^{1/y}$. If we substitute *this* equation in, we obtain:

$$({y^{1/y}})^{(y^{1/y})^{y}}=y$$ which is clearly true.

But that means...

$$x^y=y\implies x^{x^y}=y\implies x^{x^{x^y}}=y\implies x^{x^{x^{x^y}}}=y\implies \cdots\tag1$$

It is now clear that we can look at this from a different perspective: if $x^y=y$ then similarly, $y=x^y$. We can substitute this on the LHS of the former equation, therefore resulting in that $x^{x^y}=y$. Clearly, this can be done ad infinitum, and this angle of looking at it is much easier to mentally grasp.

However.... if $x^y=y$ and $x^{x^y}=y$ then this means $x^y=x^{x^y}$ which doesn't make sense. Perhaps I am missing the part that $$x^y=x^{x^y}\color{red}{\iff x^y=y}$$ for otherwise we could let $x$ and $y$ be equal to anything, even each other, in the equation $x^y=x^{x^y}$ and quickly arrive at some problems.

**Problem:**

Now I have a problem when it comes to a certain substitution. If we can carry out the implications as shown in $(1)$ forever and ever, we will get the following equation:

$$x^{x^{x^{x^{x^{\,\,\style{display: inline-block; transform: rotate(60deg)}{\vdots}}}}}} =y.\tag2$$

Considering that $\sqrt[4]{4}=\sqrt{\sqrt{4}}=\sqrt{2}$ then one can quickly deduce from *Eq.* $(2)$ that $2=4$ which is clearly wrong. Let $y=2$ then $x=\sqrt{2}$. Let $y=4$ then $x=\sqrt [4]{4}=\sqrt{2}$. Thus $2=4$.

But since we just discussed the entire concept of this equation and there doesn't seem to be any flaws with it, why am I getting the absurd flaw now that $2=4$? Clearly I did something wrong with the substitution, but that's the easy part, so I believe I am overlooking something, something likely obvious $-$ but I don't understand!

May someone please correct me on this? Any help would be much appreciated (particularly hints).

Thank you in advance.