Proving a logarithmic inequality

Question

I'm interested why this is true:

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

The following holds:

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

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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.

Answer 1

Hint

Set $$f(x,y,z)=z\frac{\ln y}{\ln x}+y\frac{\ln z}{\ln x}+x\frac{\ln z}{\ln y} $$

look at the stationary point and conclude.

Answer 2

CW answer to remove it from unanswered queue:

For $(x,y,z)=(1.1, 1.01, 1.001)$, the expression is clearly $\simeq 0.22<\frac{3}{2}$, contradicting the inequality at hand.