Answer
$\dfrac{a-b}{4}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To write the given expression, $
\ln\sqrt[4]{\dfrac{2}{3}}
,$ in terms of $a$ and $b,$ where $a=\ln2$ and $b=\ln3,$ use the laws of logarithms and substitution.
$\bf{\text{Solution Details:}}$
The given expression is equivalent to
\begin{array}{l}\require{cancel}
\ln\left( \dfrac{2}{3} \right)^{1/4}
.\end{array}
Using the Power Rule of Logarithms, which is given by $\log_b x^y=y\log_bx,$ the expression above is equivalent to:
\begin{array}{l}\require{cancel}
\dfrac{1}{4}\ln\dfrac{2}{3}
.\end{array}
Using the Quotient Rule of Logarithms, which is given by $\log_b \dfrac{x}{y}=\log_bx-\log_by,$ the expression above is equivalent to:
\begin{array}{l}\require{cancel}
\dfrac{1}{4}(\ln2-\ln3)
.\end{array}
By substitution, since $a=\ln2$ and $b=\ln3,$ the expression above is equivalent to:
\begin{array}{l}\require{cancel}
\dfrac{1}{4}(a-b)
\\\\=
\dfrac{a-b}{4}
.\end{array}