We are given two equations:

1. $x^{\log y} \cdot y^{\log x} \cdot z^{\log x} = 2$
2. $x^{\log x} \cdot y^{\log y} \cdot z^{\log z} = 6$

We are tasked with finding the value of $xyz$.

Step 1: Express both equations using logarithmic identities.

Let's simplify the first equation:

$x^{\log y} \cdot y^{\log x} \cdot z^{\log x} = 2$

Using the identity $a^{\log b} = b^{\log a}$, we can rewrite it as:

$y^{\log x} \cdot z^{\log x} \cdot x^{\log y} = 2$

Step 2: Simplify the second equation.

For the second equation:

$x^{\log x} \cdot y^{\log y} \cdot z^{\log z} = 6$

Using the same approach