compare $x^{y^{z}}$ and $y^{x^{z}}$ (help)

Question

the problem goes like this :

let $(x,y,z) \in \mathbb{R^3}_{+}$ and $0<x<y<z$ compare $ x^{y^z}$ ;$x^{z^y}$; $y^{x^z}$ ; $y^{z^x}$ ; $z^{y^x}$ ; $ z^{x^y}$ how can we compare the values ?

Answer 1

Without any additional assumptions it is impossible to compare these numbers, since for example $x^{y^z}>x^{z^y}$ for $(x,y,z)=(3,4,5)$ and $x^{y^z}<x^{z^y}$ for $(x,y,z)=(\frac{1}{2},4,5)$.

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If $a>b>e$, then $b^a>a^b$, using this fact, you can do some general work for $e<x<y<z$.