Answer
$x=6$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To solve the given equation, $
\ln e^{\ln x}-\ln(x-4)=\ln3
,$ use the laws of logarithms. Then use the properties of equality to isolate the variable.
$\bf{\text{Solution Details:}}$
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}
(\ln x)(\ln e)-\ln(x-4)=\ln3
.\end{array}
Since $\ln e=1,$ the equation above is equivalent to
\begin{array}{l}\require{cancel}
(\ln x)(1)-\ln(x-4)=\ln3
\\\\
\ln x-\ln(x-4)=\ln3
.\end{array}
Using the Quotient Rule of Logarithms, which is given by $\log_b \dfrac{x}{y}=\log_bx-\log_by,$ the equation above is equivalent
\begin{array}{l}\require{cancel}
\ln \dfrac{x}{x-4}=\ln3
.\end{array}
Since both sides have the same logarithmic base, then the logarithm can be dropped. That is,
\begin{array}{l}\require{cancel}
\dfrac{x}{x-4}=3
.\end{array}
Using the properties of equality to isolate the variable results to
\begin{array}{l}\require{cancel}
(x-4)\left( \dfrac{x}{x-4} \right)=(x-4)(3)
\\\\
x=3x-12
\\\\
x-3x=-12
\\\\
-2x=-12
\\\\
x=\dfrac{-12}{-2}
\\\\
x=6
.\end{array}
