Answer
$x=-3$
Work Step by Step
Using the properties of logarithms, the given equation, $
\log 4x-\log (x-3)=\log 2
$, is equivalent to
\begin{align*}\require{cancel}
\log \dfrac{4x}{x-3}&=\log 2
&(\text{use }\log_b \dfrac{x}{y}=\log_b x-\log_b y)
.\end{align*}
Since $\log_b m=\log_b n $ implies $m=n$, the equation above implies
\begin{align*}\require{cancel}
\dfrac{4x}{x-3}&=2
.\end{align*}
Using the properties of equality, the equation above is equivalent to
\begin{align*}\require{cancel}
(\cancel{x-3})\cdot\dfrac{4x}{\cancel{x-3}}&=2\cdot(x-3)
\\\\
4x&=2x-6
\\
4x-2x&=-6
\\
2x&=-6
\\\\
\dfrac{\cancel2x}{\cancel2}&=\dfrac{-6}{2}
\\\\
x&=-3
.\end{align*}
Hence, the solution to the equation $
\log 4x-\log (x-3)=\log 2
$ is $
x=-3
$.
