Answer
$r=\pm\dfrac{\sqrt{e^3}}{2}$
Work Step by Step
Recall:
$$\ln{a}=y \longleftrightarrow e^y=a$$
Use the definition above to obtain:
\begin{align*}
\ln{4r^2}&=3\\\\
e^{3}&=4r^2\\\\
\frac{e^3}{4}&=\frac{4r^2}{4}\\\\
\frac{e^3}{4}&=r^2\\\\
\pm \sqrt{\frac{e^3}{4}}&=r\\\\
\pm\frac{\sqrt{e^3}}{2}&=r
\end{align*}
Check:
\begin{align*}
\ln{\left(4\left(\frac{\sqrt{e^3}}{2}\right)^2\right)}&\stackrel{?}=3\\\\
\ln{\left(4\left(\frac{e^3}{4}\right)\right)}&\stackrel{?}=3\\\\
\ln{e^3}&\stackrel{?}=3\\\\
3\stackrel{\checkmark}=3
\end{align*}
\begin{align*}
\ln{\left(4\left(-\frac{\sqrt{e^3}}{2}\right)^2\right)}&\stackrel{?}=3\\\\
\ln{\left(4\left(\frac{e^3}{4}\right)\right)}&\stackrel{?}=3\\\\
\ln{e^3}&\stackrel{?}=3\\\\
3\stackrel{\checkmark}=3
\end{align*}
