Answer
$x\approx 9.386$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To solve the given equation, $
e^{3x-7}\cdot e^{-2x}=4e
,$ use the laws of exponents to combine the factors with the same base. Then take the natural logarithm of both sides and use the properties of logarithms. Finally, use the properties of equality to isolate the variable.
$\bf{\text{Solution Details:}}$
Using the Product Rule of the laws of exponents which is given by $x^m\cdot x^n=x^{m+n},$ the equation above is equivalent to
\begin{array}{l}\require{cancel}
e^{3x-7+(-2x)}=4e
\\\\
e^{3x-7-2x}=4e
\\\\
e^{x-7}=4e
.\end{array}
Taking the natural logarithm of both sides, the equation above is equivalent to
\begin{array}{l}\require{cancel}
\ln e^{x-7}=\ln (4e)
.\end{array}
Using the Power Rule of Logarithms, which is given by $\log_b x^y=y\log_bx,$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
(x-7)\ln e=\ln (4e)
.\end{array}
Using the Product Rule of Logarithms, which is given by $\log_b (xy)=\log_bx+\log_by,$ the expression above is equivalent
\begin{array}{l}\require{cancel}
(x-7)\ln e=\ln 4+\ln e
.\end{array}
Since $\ln e=1$, the equation above is equivalent to
\begin{array}{l}\require{cancel}
(x-7)(1)=\ln 4+1
\\\\
x-7=\ln 4+1
.\end{array}
Using the properties of equality to isolate the variable, the equation above is equivalent to
\begin{array}{l}\require{cancel}
x=\ln 4+1+7
\\\\
x=\ln 4+8
\\\\
x\approx 9.386
.\end{array}
