[tex]\log_{ \sqrt{2} } \left(\cfrac{ \sqrt[5]{4} }{8\cdot \sqrt{2} }\right) \\ \\ \\ = \log_{2^{ \frac{1}{2} }} \left(\cfrac{ 2^{ \frac{2}{5} }}{2^3\cdot 2^{ \frac{1}{2} } }\right) \\ \\ \\ = \log_{2^{ \frac{1}{2} } }\left( \cfrac{ 2^{ \frac{2}{5} }}{2^{ \frac{7}{2} } }\right) \\ \\ \\ = \log_{2^{ \frac{1}{2} } }\left(2^{- \frac{31}{10} } \right) \\\\\\ = - \cfrac{31}{10}: \cfrac{1}{2} \cdot \log_2(2) \\\\\\ =- \cfrac{31}{5} \cdot1 \\\\\\ =- \cfrac{31}{5}[/tex]
