#### Answer

$x\approx-0.123$

#### Work Step by Step

$\bf{\text{Solution Outline:}}$
To solve the given equation, $
2e^{5x+2}=8
,$ use the properties of equality to isolate the $e$ expression. Then take the natural logarithm of both sides. Use the laws of logarithms and the properties of equality to isolate the variable. Finally, Express the answer with $3$ decimal places.
$\bf{\text{Solution Details:}}$
Using the properties of equality to isolate the $e$ expression results to
\begin{array}{l}\require{cancel}
\dfrac{2e^{5x+2}}{2}=\dfrac{8}{2}
\\\\
e^{5x+2}=4
.\end{array}
Taking the natural logarithm of both sides, the equation above is equivalent to
\begin{array}{l}\require{cancel}
\ln e^{5x+2}=\ln4
.\end{array}
Using the Power Rule of Logarithms, which is given by $\log_b x^y=y\log_bx,$ the equation above is equivalent
\begin{array}{l}\require{cancel}
(5x+2)\ln e=\ln4
.\end{array}
Since $\ln e =1,$ the equation above is equivalent to
\begin{array}{l}\require{cancel}
(5x+2)(1)=\ln4
\\\\
5x+2=\ln4
.\end{array}
Using the properties of equality to isolate the variable results to
\begin{array}{l}\require{cancel}
5x=-2+\ln4
\\\\
x=\dfrac{-2+\ln4}{5}
\\\\
x\approx-0.123
.\end{array}