HMMT November 2023 U13

Suppose $x, y$, and $z$ are real numbers greater than 1 such that

$$\begin{aligned} & x^{\log _{y} z}=2 \\ & y^{\log _{z} x}=4, \text { and } \\ & z^{\log _{x} y}=8 \end{aligned}$$

Compute $\log _{x} y$.