Proving a logarithmic inequality

I’m interested why this is true:

$$ \text{Considering }\forall (x,y,z) \in (1,\infty) $$

The following holds:

$$\log_xy^z+\log_x{z^y}+log_y{z^x} \geq \frac{3}{2}$$

This is taken from a high school textbook of mine. I tried finding a meaningful manipulation by using AM-GM, but that got pretty messy. I’d like to avoid Lagrange multipliers since this is meant to be a pretty basic problem.

Any progress would be appreciated.

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