Answer
$x=\dfrac{43}{21}$
Work Step by Step
Using the properties of logarithms, the given equation, $
\log_4(x+1)-\log_4(x-2)=3
,$ is equivalent to
\begin{array}{l}\require{cancel}
\log_4\dfrac{x+1}{x-2}=3
.\end{array}
Since $y=b^x$ is equivalent to $\log_by=x$, the equation aboveis equivalent to
\begin{array}{l}\require{cancel}
\dfrac{x+1}{x-2}=4^3
.\end{array}
Using the properties of equality, the solution to the equation, $
\dfrac{x+1}{x-2}=4^3
,$ is
\begin{array}{l}\require{cancel}
\dfrac{x+1}{x-2}=64
\\\\
x+1=64(x-2)
\\\\
x+1=64x-128
\\\\
x-64x=-128-1
\\\\
-63x=-129
\\\\
x=\dfrac{-129}{-63}
\\\\
x=\dfrac{129}{63}
\\\\
x=\dfrac{\cancel{3}\cdot43}{\cancel{3}\cdot21}
\\\\
x=\dfrac{43}{21}
.\end{array}
