If possible, use logarithm properties to rewrite the expression $\frac{\log \sqrt{x^{3}}}{\log \frac{y}{z^{6}}}$, in terms of $u, v, w$ given that $u=\log x, v=\log y$, $w=\log z$.
NOTE: Your answer should not involve logs.
\[
\frac{\log \sqrt{x^{3}}}{\log \frac{y}{z^{6}}}=
\]
Answer
\(\boxed{\frac{\log \sqrt{x^{3}}}{\log \frac{y}{z^{6}}} = \frac{\frac{3}{2}u}{v - 6w}}\)
Step 1 ：Rewrite the numerator \(\log \sqrt{x^{3}}\) as \(\frac{3}{2} \log x = \frac{3}{2}u\).
Step 2 ：Rewrite the denominator \(\log \frac{y}{z^{6}}\) as \(\log y - 6 \log z = v - 6w\).
Step 3 ：Substitute these into the original expression to get \(\frac{\log \sqrt{x^{3}}}{\log \frac{y}{z^{6}}} = \frac{\frac{3}{2}u}{v - 6w}\).
Step 4 ：\(\boxed{\frac{\log \sqrt{x^{3}}}{\log \frac{y}{z^{6}}} = \frac{\frac{3}{2}u}{v - 6w}}\)
