Answer
$x=\dfrac{192}{127}$
Work Step by Step
Using the properties of logarithms, the given expression, $
\log_4 x-\log_4 (2x-3)=3
,$ is equivalent to
\begin{array}{l}\require{cancel}
\log_4 \dfrac{x}{2x-3}=3
.\end{array}
Since $y=b^x$ is equivalent to $\log_b y=x$, then the solution to the equation, $
\log_4 \dfrac{x}{2x-3}=3
,$ is
\begin{array}{l}\require{cancel}
\dfrac{x}{2x-3}=4^3
\\\\
\dfrac{x}{2x-3}=64
\\\\
x=64(2x-3)
\\\\
x=128x-192
\\\\
x-128x=-192
\\\\
-127x=-192
\\\\
x=\dfrac{-192}{-127}
\\\\
x=\dfrac{192}{127}
.\end{array}
Upon checking, $
x=\dfrac{192}{127}
$ satisfies the original equation.
