Question 3: Expand the following using laws of logarithms:
\((v)\) \(\quad \) \(\ln\sqrt[3]{16x^{3}} \)
Solution:
\[
\begin{align*}
&=\ln\sqrt[3]{16x^{3}}\\
&=\ln\sqrt[3]{2^{4} x^{3}}\\
&=\ln\left( 2^{4} x^{3}\right)^{\large{\frac{1}{3}}}\\
&=\ln\left( 2^{\large{\frac{4}{3}}} x^{\large{\frac{3}{3}}}\right)\\
&=\ln\left( 2^{\large{\frac{4}{3}}} x\right)\\
&=\ln 2^{\large{\frac{4}{3}}} +\ln x\\
&=\frac{4}{3}\ln 2 +\ln x\\
\end{align*}
\]