Problem

This problem rewards substitution: let $a = \log_{10} x$, $b = \log_{10} y$, and $c = \log_{10} z$ to convert the logarithmic system into polynomial equations in $a$, $b$, and $c$. Once you expand each equation using log properties, you'll get three equations that are linear or quadratic in your new variables—much easier to handle than the original form. The key is recognizing that the system is actually hiding a simpler algebraic structure underneath the logarithms.

This problem rewards substitution: let $a = \log_{10} x$, $b = \log_{10} y$, and $c = \log_{10} z$ to convert the logarithmic system into polynomial equations in $a$, $b$, and $c$. Once you expand each equation using log properties, you'll get three equations that are linear or quadratic in your new variables—much easier to handle than the original form. The key is recognizing that the system is actually hiding a simpler algebraic structure underneath the logarithms.