Answer
$\dfrac{1}{3}u+4v$
Work Step by Step
$\bf{\text{Solution Outline:}}$
Use the properties of logarithms to express the given expression, $
\ln \left(\sqrt[3]{a}\cdot b^4\right)
,$ in a form that uses $\ln a$ and $\ln b$ only. Then substitute $u$ for any instance of $\ln a$ and substitute $v$ for any instance of $\ln b.$
$\bf{\text{Solution Details:}}$
Using the Product Rule of Logarithms, which is given by $\log_b (xy)=\log_bx+\log_by,$ the expression above is equivalent
\begin{array}{l}\require{cancel}
\ln \sqrt[3]{a}+\ln b^4
.\end{array}
Using the definition of rational exponents which is given by $a^{\frac{m}{n}}=\sqrt[n]{a^m}=\left(\sqrt[n]{a}\right)^m,$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
\ln a^{1/3}+\ln b^4
.\end{array}
Using the Power Rule of Logarithms, which is given by $\log_b x^y=y\log_bx,$ the expression above is equivalent
\begin{array}{l}\require{cancel}
\dfrac{1}{3}\ln a+4\ln b
.\end{array}
Substituting $u$ with $\ln a$ and $v$ with $\ln b,$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
\dfrac{1}{3}u+4v
.\end{array}
