Answer
$x=-309,\text{ }316$
Work Step by Step
Since $\log_b y=x$ implies $b^x=y,$ the given equation, $
\log_5 |2x-7|=4,$ is equivalent to
\begin{align*}\require{cancel}
5^4&=|2x-7|
\\
625&=|2x-7|
\\
|2x-7|&=625
.\end{align*}
Since $|x|=c$ ($c$ is a positive constant) implies $x=c\text{ or }x=-c,$ the equation above is equivalent to
\begin{align*}
2x-7&=625
\text{ or }2x-7=-625
.\end{align*}
Using the properties of equality, the solutions to the equations above are
\begin{array}{l|r}
2x-7=625 & 2x-7=-625
\\
2x=625+7 & 2x=-625+7
\\
2x=632 & 2x=-618
\\\\
\dfrac{\cancel2x}{\cancel2}=\dfrac{632}{2} & \dfrac{\cancel2x}{\cancel2}=-\dfrac{618}{2}
\\\\
x=316 & x=-309
.\end{array}
Hence, the solutions to $\log_5 |2x-7|=4$ are $x=-309,\text{ }316$.
