Answer
$x\approx5.879$
Work Step by Step
Using the properties of logarithms, the given equation, $
\ln e^{0.45x}=\sqrt{7}
$ is equivalent to
\begin{align*}\require{cancel}
0.45x\ln e&=\sqrt{7}
&(\text{use }\log_b x^y=y\log_b x)
\\
0.45x(1)&=\sqrt{7}
&(\text{use }\ln e=1)
\\
0.45x&=\sqrt{7}
.\end{align*}
Using the properties of equality, the equation above is equivalent to
\begin{align*}\require{cancel}
\dfrac{\cancel{0.45}x}{\cancel{0.45}}&=\dfrac{\sqrt{7}}{0.45}
\\\\
x&=\dfrac{\sqrt{7}}{0.45}
\\\\
x&\approx5.879
.\end{align*}
Hence, the solution to the equation $
\ln e^{0.45x}=\sqrt{7}
$ is $
x\approx5.879
$.