#### Answer

$x\approx2.102$

#### Work Step by Step

$\bf{\text{Solution Outline:}}$
To solve the given equation, $
10e^{3x-7}=5
,$ 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{10e^{3x-7}}{10}=\dfrac{5}{10}
\\\\
e^{3x-7}=\dfrac{1}{2}
.\end{array}
Taking the natural logarithm of both sides, the equation above is equivalent to
\begin{array}{l}\require{cancel}
\ln e^{3x-7}=\ln \dfrac{1}{2}
.\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}
(3x-7)\ln e=\ln \dfrac{1}{2}
.\end{array}
Since $\ln e =1,$ the equation above is equivalent to
\begin{array}{l}\require{cancel}
(3x-7)(1)=\ln \dfrac{1}{2}
\\\\
3x-7=\ln \dfrac{1}{2}
.\end{array}
Using the Quotient Rule of Logarithms, which is given by $\log_b \dfrac{x}{y}=\log_bx-\log_by,$ the expression above is equivalent
\begin{array}{l}\require{cancel}
3x-7=\ln 1-\ln2
.\end{array}
Since $\ln1=0,$ the equation above is equivalent to
\begin{array}{l}\require{cancel}
3x-7=0-\ln2
\\\\
3x-7=-\ln2
.\end{array}
Using the properties of equality to isolate the variable results to
\begin{array}{l}\require{cancel}
3x=7-\ln2
\\\\
x=\dfrac{7-\ln2}{3}
\\\\
x\approx2.102
.\end{array}