Question

Show that $\log \frac{\sqrt[4]{45}.\sqrt[10]{102}}{\sqrt[3]{318.\sqrt{2}}}=\frac{1}{4}\log 5-\frac{2}{5}\log 2-\frac{2}{3}\log 3$.

Answer 1

$\log \frac{\sqrt[4]{45}.\sqrt[10]{102}}{\sqrt[3]{318.\sqrt{2}}}=\log \frac{5^{\frac{1}{4}}.2^{\frac{1}{10}}}{\sqrt[3]{33^2.2^{\frac{3}{2}}}}=\log \frac{5^{\frac{1}{4}}.2^{\frac{1}{10}}}{3^{\frac{2}{3}}.2^{\frac{1}{2}}}=\log \frac{5^{\frac{1}{4}}.2^{\frac{-2}{5}}}{3^{\frac{2}{3}}}=\frac{1}{4}\log 5-\frac{2}{5}\log 2-\frac{2}{3}\log 3$