Answer
$x=\dfrac{e^2\sqrt{2e}}{4}\approx 4.3072$
Work Step by Step
\begin{align*}
\ln{x^2}+\ln{2^3}&=5 &\text{Power Property of Logarithms}\\\\
\ln{x^2}+\ln8&=5 &\text{Write the equation in exponential form.}\\\\
\ln{(x^2\cdot8)}&=5 &\text{Product Property of Logarithms}\\\\
\ln{(8x^2)}&=5 &\text{Product Property of Logarithms}\\\\
e^5&=8x^2&\text{Write the equation in exponential form.}\\\\
\frac{e^5}{8}&=\frac{8x^2}{8}&\text{Divide 8 to both sides.}\\\\
\frac{e^5}{8}&=x^2\\\\
\pm\sqrt{\frac{e^5}{8}}&=\sqrt{x^2} &\text{Take the square root of both sides.}\\\\
\pm\sqrt{\frac{e^5}{8}\cdot \frac{2}{2}}&=x &\text{Simplify.}\\\\
\pm\sqrt{\frac{2e^5}{16}}&=x \\\\
\pm\frac{e^2}{4}\sqrt{2e}&=x \\\\
x&=\pm \frac{e^2\sqrt{2e}}{4}\\\\
x&\approx \pm 4.3072 &\text{Evaluate using a calculator.}
\end{align*}
Note that in $\ln{x}$, $x$ has to be greater than $0$.
This means that in the given equation, $x\approx -4.3072$ is an extraneous solution.
Check:
\begin{align*}
2\ln{\left(\frac{e^2\sqrt{2e}}{4}\right)}+3\ln{\left(\frac{e^2\sqrt{2e}}{4}\right)}&\stackrel{?}=5\\\\
5&=5
\end{align*}
Thus, the solution is $x=\dfrac{e^2\sqrt{2e}}{4}\approx 4.3072$.
