Finding the result of $xyz$

Question

$$2^x = 3$$ $$5^y = 2$$ $$3^z = 125$$

Find the result of $xyz$

To get $xyz$, I've tried to multiply all together.

$$2^x . 5^y . 3^z = 750$$

Unfortunalety, I've gone too wrong as in my perspective. Can you assist? I'd like to get your professional tips.

EDIT: I'm trying to solve this question by using exponential properties.

Answer 1

$$5^{xyz}=(5^y)^{xz}=2^{xz}=(2^x)^z=3^z=125$$

Answer 2

Solve for each $x=\log_2(3)$, $y=\log_5(2)$ and $z=\log_3(125)$.

The product would be $xyz=\log_2(3)\log_5(2)\log_3(125)$.

Very likely they will examine also the possibility of some simplification.

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For the new requirement:

Raise the first equation to $z$. You get $2^{xz}=3^z=125=5^3=2^{3/y}$.

Raise again, but to $y$ and get $2^{xyz}=2^3$.

Answer 3

taking the logarithm we get $$x=\frac{\ln(3)}{\ln(2)}$$ $$y=\frac{\ln(2)}{\ln(5)}$$ $$z=\frac{\ln(125)}{\ln(3)}$$ Can you proceed?