Answer
$\text{no solution}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To solve the given equation, $
\ln x-4\ln 3=\ln\left( \dfrac{1}{5} x \right)
,$ use the properties of logarithms to simplify the expression at the left. Then drop the logarithm on both sides and 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 3^4=\ln\left( \dfrac{1}{5} x \right)
\\\\
\ln x-\ln 81=\ln\left( \dfrac{1}{5} x \right)
.\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}{81}=\ln\left( \dfrac{1}{5} x \right)
.\end{array}
Since both sides use the same logarithmic base, then the logarithm can be dropped. Hence, the equation above is equivalent to
\begin{array}{l}\require{cancel}
\dfrac{x}{81}=\dfrac{x}{5}
.\end{array}
Since $\dfrac{a}{b}=\dfrac{c}{d}$ implies $ad=bc$ or sometimes referred to as cross-multiplication, the equation above is equivalent to
\begin{array}{l}\require{cancel}
x(5)=81(x)
\\\\
5x=81x
\\\\
5x-81x=0
\\\\
-76x=0
\\\\
x=0
.\end{array}
If $x=0,$ the part of the given equation, $
\ln x
,$ becomes $\ln 0$ which is not a real number. Hence, there is $\text{
no solution
.}$
