Answer
$x\approx-0.693$
Work Step by Step
Using the properties of equality, the given equation, $
7+3e^{-x}=13
,$ is equivalent to
\begin{array}{l}\require{cancel}
3e^{-x}=13-7
\\\\
3e^{-x}=6
\\\\
e^{-x}=\dfrac{6}{3}
\\\\
e^{-x}=2
.\end{array}
Taking the natural logarithm of both sides and using the properties of logarithms, the value of the variable that satisfies the equation, $
e^{-x}=2
,$ is
\begin{array}{l}\require{cancel}
\ln e^{-x}=\ln 2
\\\\
-x(\ln e)=\ln 2
\\\\
-x(1)=\ln 2
\\\\
-x=\ln 2
\\\\
x=-\ln 2
\\\\
x\approx-0.693
.\end{array}
