Let $0<a<1$, for which $x>0$ is $x^{a^x} = a^{x^a}$ true.

Question

Let $0<a<1$, for which $x>0$ is $x^{a^x} = a^{x^a}$ true.

This is where I have gotten so far:

$\log_a( x^{a^x} )= \log_a( a^{x^a} ) $

$ a^x \log_ax = x^a $

Now, I know I am only considering positive x, but I am not sure where to go from here. I searched the site and cannot find any close examples either.

Answer 1

Hint

For $x \gt 0$, $y = x^{a^x}$ in an increasing function and $y = a^{x^a}$ is a decreasing function. Hence the only solution for $x \gt 0$ is the trivial solution, i.e. $\color{blue}{x = a}$. It is not difficult to prove the increasing and decreasing nature of these functions.