Question

Find the logarithms of
$\sqrt[4]{4a^{\frac{8}{5}}},\frac{1}{a^{\frac{1}{2}}},\sqrt[3]{3a^{-\frac{15}{2}}}$ to base $a$.

Answer 1

${\log }_a\sqrt[4]{4a^{\frac{8}{5}}}={\log }_a{(a}^{\frac{8}{5}}{)}^{\frac{1}{4}}={\log }_a{(a}^{\frac{2}{5}})=\frac{2}{5}$

${\log }_a\frac{1}{a^{\frac{1}{2}}}={\log }_a{(a}^{\frac{-1}{2}})=\frac{-1}{2}{\log }_a(a)=-\frac{1}{2}$

${\log }_a\sqrt[3]{3a^{\frac{-15}{2}}}={\log }_a{(a}^{\frac{-15}{2}.\frac{1}{3}})={\log }_a(a^{\frac{-5}{2}})=-\frac{5}{2}$