How many triplets (x, y, z) of positive real numbers can be found such that $x^{y} = z, y^{z} = x a n d z^{x} = y$ ?

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Solution

$x^{y} = z$ , $y^{z} = x$ and $z^{x} = y$
Put $x = y^{z}$ in $(x^{y} = z)$
$\Rightarrow y^{z y} = z,$
Now put $y = z^{x}$
$z^{x y z} = z$ $\Rightarrow x y z = 1$
Hence only one value of $(x, y, z)$ is possible.

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