Exponential Dilemma

By the substitution of $k = y^{z}$ , we reduce this to the form

$x^k = k^x$

Which, by all means, will make it clear that $x = k$ makes the statement true (well, uhm, that is aside from the two other solutions, i.e., $x=2, k=4$ and $x=4, k=2$ ) . This will mean that for all $x = n^n$ for some integer $n$ , and this will give us an infinite set of triples of the form $(n^n, n, n)$ for all positive integers $n>2$ . There.

An infinite family is given by $(x,y,z) = (1,-1,2n)$ , where $n$ is an integer.