\[ \large \color{green}x \color{brown}y=10^a, \quad \color{brown}y \color{red}z=10^b, \quad \color{green}x \color{red}z=10^c \]

Consider the system of equations above, what is \(\log \ \color{green} x +\log \ \color{brown}y+ \log \ \color{red}z\)?

**Details and Assumptions**:

The logarithms have base of \(10\)

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