Answer
$-4$
Work Step by Step
$\bf{\text{Solution Outline:}}$
Use the properties of logarithms to evaluate the given expression, $
\ln\left( \dfrac{1}{e^4} \right)
.$
$\bf{\text{Solution Details:}}$
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}
\ln1-\ln e^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}
\ln1-4\ln e
.\end{array}
Since $\ln e=1$ and $\ln1=0,$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
0-4(1)
\\\\=
-4
.\end{array}