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I stumbled across this problem:

$x^{x^{x^{...}}}=2$

Obviously, I used the substitution trick and I got

$x^2=2$

and thus, $x=\pm\sqrt{2}$. I have tested that this works.

However, I tried to solve

$x^{x^{x^{...}}}=4$

and it yields the same real answers ($x^4=4$). I have no idea why this is.

Jed
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  • Why don't you compute the power tower for those two values ? – mercio Dec 31 '15 at 02:40
  • $\pm\sqrt{2}$ don't work for the second tower, but they work for the first one. Why this is the case is what I'm trying to figure out. – Jed Dec 31 '15 at 02:40
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    Note that in the first example, $-\sqrt{2}$ also does not work. Only $\sqrt{2}$ is the solution. – Fabian Dec 31 '15 at 02:49
  • The second example does not have a solution as can be checked by trying the four candidates $\pm \sqrt{2}, \pm i \sqrt{2}$. – Fabian Dec 31 '15 at 02:50
  • How did you compute the tower with $-\sqrt 2$ ?? – mercio Dec 31 '15 at 02:52
  • @mercio: I did not calculate it. For $x$ that is not positive you run into the problem of the multivaluedness of the complex logarithm. So you have to choose a principle value. Whatever you do, the result will be definitely not real ;-) I just wanted to point out to the OP that $-\sqrt{2}$ also does not work in the first example. – Fabian Dec 31 '15 at 02:58

2 Answers2

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Note that $x^{x^{x^{\vdots}}} = 4$ implies that $x^4 = 4$, but not the other way around.

So any solution to the first equation must be a solution to the second equation. However, not all solutions to the second equation will be solutions to the first equation, i.e. the second equation may have some extraneous solutions.

Note: This is similar to what is going on in this other question.

Community
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JimmyK4542
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The infinite tetration $f(x)=x^{x^{x^\cdots}}$ only converges for $(1/e)^e \leq x \leq e^{e-1}$ and assumes values in $[1/e,e]$. Thus the inverse function $f^{-1}(y)$ is only defined for $y\in [1/e,e]$. So the solution of the first example is $\sqrt{2}$ whereas the second example does not have a solution (since $4>e$).

Fabian
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