Answer
$x=9$
Work Step by Step
Using the properties of logarithms, the given expression, $
\log_3 x+\log_3 (x-8)=2
,$ is equivalent to
\begin{array}{l}\require{cancel}
\log_3 [x(x-8)]=2
\\\\
\log_3 (x^2-8x)=2
.\end{array}
Since $y=b^x$ is equivalent to $\log_b y=x$, then the solution to the equation, $
\log_3 (x^2-8x)=2
,$ is
\begin{array}{l}\require{cancel}
x^2-8x=3^2
\\\\
x^2-8x=9
\\\\
x^2-8x-9=0
\\\\
(x-9)(x+1)=0
\\\\
x=\left\{ -1,9 \right\}
.\end{array}
Upon checking, only $
x=9
$ satisfies the original equation.
