Answer
$3u-2v$
Work Step by Step
$\bf{\text{Solution Outline:}}$
Use the properties of logarithms to express the given expression, $
\ln \dfrac{a^3}{b^2}
,$ 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 Quotient Rule of Logarithms, which is given by $\log_b \dfrac{x}{y}=\log_bx-\log_by,$ the expression above is equivalent
\begin{array}{l}\require{cancel}
\ln a^3-\ln b^2
.\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}
3\ln a-2\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}
3u-2v
.\end{array}
