Answer
$\dfrac{a+b}{5}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To write the given expression, $
\ln\sqrt[5]{6}
,$ 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\sqrt[5]{2\cdot3}
\\\\=
\ln(2\cdot3)^{1/5}
.\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}{5}\ln(2\cdot3)
.\end{array}
Using the Product Rule of Logarithms, which is given by $\log_b (xy)=\log_bx+\log_by,$ the expression above is equivalent to:
\begin{array}{l}\require{cancel}
\dfrac{1}{5}(\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}{5}(a+b)
\\\\=
\dfrac{a+b}{5}
.\end{array}