\(\frac{\log \sqrt{27}}{\log 81}\) = \(\frac{\log \sqrt{3^3}}{\log 3^4}\) = \(\frac{\log 3^{\frac{3}{2}}}{\log 3^4}\) = \(\frac{\frac{3}{2} \log 3}{4 \log 3}\) = \(\frac{\frac{3}{2}}{4}\) = \(\frac{3}{8}\)