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 :\(\boxed{\frac{\log \sqrt{x^{3}}}{\log \frac{y}{z^{6}}} = \frac{\frac{3}{2}u}{v - 6w}}\)