Answer
$x=\dfrac{e^3}{2}$
Work Step by Step
Using the properties of the logarithms, the given equation, $
\ln2x=3
,$ is equivalent to
\begin{align*}\require{cancel}
\log_e2x&=3
&\left(\text{use } \ln x=\log_e x \right)
\\
2x&=e^3
&\left(\text{use } y=b^x\Rightarrow\log_by=x \right)
\\\\
\dfrac{2x}{2}&=\dfrac{e^3}{2}
\\\\
x&=\dfrac{e^3}{2}
.\end{align*}
Hence, the solution is $
x=\dfrac{e^3}{2}
.$
CHECKING: Substituting $x=\dfrac{e^3}{2}$ in the given equation, $\ln2x=3,$ results to
\begin{align*}
\ln\left[2\left(\dfrac{e^3}{2}\right)\right]&=3
\\\\
\ln\left[\cancel2\left(\dfrac{e^3}{\cancel2}\right)\right]&=3
\\\\
\ln e^3&=3
\\
3(\ln e)&=3
&\left(\text{use } \ln a^x=x(\ln a) \right)
\\
3(1)&=3
&\left(\text{use } \ln e=1 \right)
\\
3&=3
&\text{ (TRUE)}
.\end{align*}
Since the substitution above ended with a TRUE statement, then $
x=\dfrac{e^3}{2}
$ is the solution.