Answer
$x=\dfrac{4}{3}$
Work Step by Step
Using the properties of logarithms, the given equation, $
\log 5x-\log (2x-1)=\log4
$, is equivalent to
\begin{align*}\require{cancel}
\log \dfrac{5x}{2x-1}&=\log4
&(\text{use }\log_b x^y=y\log_b x)
.\end{align*}
Since $\log_b x=\log_b y$ implies $x=y$, then the equation above implies
\begin{align*}\require{cancel}
\dfrac{5x}{2x-1}&=4
.\end{align*}
Using the properties of equality, the equation above is equivalent to
\begin{align*}\require{cancel}
(\cancel{2x-1})\cdot\dfrac{5x}{\cancel{2x-1}}&=4\cdot(2x-1)
\\\\
5x&=8x-4
\\
4&=8x-5x
\\
4&=3x
\\\\
\dfrac{4}{3}&=\dfrac{\cancel3x}{\cancel3}
\\\\
\dfrac{4}{3}&=x
.\end{align*}
Hence, the solution to the equation $
\log 5x-\log (2x-1)=\log4
$ is $
x=\dfrac{4}{3}
$.
