[tex]\displaystyle \mathtt{log_{ \frac{1}{5} } \sqrt[3]{ \frac{
\sqrt[4]{125 \sqrt{5} } }{ 5\sqrt[4]{25 \sqrt{5} } } } =log_{ \frac{1}{5} }
\sqrt[3]{ \frac{ \sqrt[4]{5^3 \cdot 5^{ \frac{1}{2} }} }{5 \sqrt[4]{5^2\cdot
5^{ \frac{1}{2} }} } } =log_{ \frac{1}{5} } \sqrt[3]{ \frac{ \sqrt[4]{5^{3+
\frac{1}{2} }} }{5 \sqrt[4]{5^{2+ \frac{1}{2} }} } }= }[/tex]
[tex]\displaystyle \mathtt{=log_{ \frac{1}{5} }
\sqrt[3]{ \frac{ \sqrt[4]{5^{ \frac{7}{2} }} }{5 \sqrt[4]{5^{ \frac{5}{2} }} }
}=log_{ \frac{1}{5} } \sqrt[3]{ \frac{5^{ \frac{ \frac{7}{2} }{4} }}{5 \cdot
5^{ \frac{ \frac{5}{2} }{4} }} } =log_{ \frac{1}{5} } \sqrt[3]{ \frac{5^{
\frac{7}{2}\cdot \frac{1}{4} }}{5 \cdot 5^{ \frac{5}{2}\cdot \frac{1}{4} }} }
=log_{ \frac{1}{5} } \sqrt[3]{ \frac{5^{ \frac{7}{8} }}{5 \cdot 5^{ \frac{5}{8}
}} } =}[/tex]
[tex]\displaystyle \mathtt{=log_{ \frac{1}{5} }
\sqrt[3]{ \frac{5^{ \frac{7}{8} }}{5^{1+ \frac{5}{8} }} }=log_{ \frac{1}{5} }
\sqrt[3]{ \frac{5^{ \frac{7}{8} }}{5^{ \frac{13}{8} }} } =log_{ \frac{1}{5} }
\frac{ \sqrt[3]{5^{ \frac{7}{8} }} }{ \sqrt[3]{5^{ \frac{13}{8} }} } =log_{
\frac{1}{5} } \frac{5^{ \frac{ \frac{7}{8} }{3} }}{5^{ \frac{ \frac{13}{8} }{3}
}}= } [/tex]
[tex]\displaystyle \mathtt{=log_{ \frac{1}{5} }
\frac{5^{ \frac{7}{8}\cdot \frac{1}{3} }}{5^{ \frac{13}{8}\cdot \frac{1}{3} }}
=log_{ \frac{1}{5} } \frac{5^{ \frac{7}{24} }}{5^{ \frac{13}{24} }}=log_{
\frac{1}{5} } 5^{ \frac{7}{24}- \frac{13}{24} }=log_{ \frac{1}{5} }5^{-
\frac{6}{24} }=log_{5^{-1}} 5^{- \frac{1}{4} }=} \\ \\ \mathtt{= \frac{1}{4}
log_{5}5= \frac{1}{4} }[/tex]
