Answer
$x=\dfrac{4}{e^2}\approx 0.5413$
Work Step by Step
\begin{align*}
\ln{\frac{4}{x}}=2 &\text{Quotient Property of Logarithms}\\\\
e^2&=\frac{4}{x}&\text{Write the equation in exponential form.}\\\\
x(e^2)&=4&\text{Cross-multiply.}\\\\
\frac{x(e^2)}{e^2}&=\frac{4}{e^2} &\text{Divide $e^2$ to both sides.}\\\\
x&=\frac{4}{e^2}\\\\
x&\approx 0.5413\end{align*}
Check:
\begin{align*}
\ln4 -\ln{\frac{4}{e^2}}&\stackrel{?}=2\\\\
2&\stackrel{\checkmark}=2\end{align*}
Thus, the solution is $x=\dfrac{4}{e^2}\approx 0.5413$.
