Question
# Express the following logarithms in terms of $\log a$ and $\log b$ .$\log {\left( {\sqrt {{a^2}{b^3}} } \right)^6}$
$\Rightarrow \log {\left( {\sqrt {{a^2}{b^3}} } \right)^6} \\ \Rightarrow 6\log \left( {\sqrt {{a^2}{b^3}} } \right){\text{ }}\left[ {\because \log {m^n} = n\log m} \right] \\ \Rightarrow 6\log {\left( {{a^2}{b^3}} \right)^{\dfrac{1}{2}}} \\ \Rightarrow \dfrac{6}{2}\log \left( {{a^2}{b^3}} \right) \\ \Rightarrow 3\log \left( {{a^2}{b^3}} \right) \\ \Rightarrow 3\left( {\log \left( {{a^2}} \right) + \log \left( {{b^3}} \right)} \right){\text{ }}\left[ {\because \log mn = \log m + \log n} \right] \\ \Rightarrow 3\left( {2\log a + 3\log b} \right){\text{ }}\left[ {\because \log {m^n} = n\log m} \right] \\ \Rightarrow 6\log a + 9\log b \\$