Answer
The solution set is$\left\{ 9 \right\}$.
Work Step by Step
A logarithmic equation in one variable can be solved when the equation is written in exponential form.
For example, the equation \[{{\log }_{a}}x=b\] when written in exponential form is\[x={{a}^{b}}\].
Hence, solution for x is \[x={{a}^{b}}\].
Now, for the provided logarithmic equation, simplify by logarithmic properties:
\[\begin{align}
& {{\log }_{9}}\left( x \right)+{{\log }_{9}}\left( x-8 \right)=1 \\
& {{\log }_{9}}\left( x\left( x-8 \right) \right)=1
\end{align}\]
Now, rewrite the logarithmic equation in the exponential form:
\[\begin{align}
& x\left( x-8 \right)=9 \\
& {{x}^{2}}-8x-9=0 \\
& {{x}^{2}}-9x+x-9=0 \\
& x\left( x-9 \right)+1\left( x-9 \right)=0
\end{align}\]
Solve further as:
\[\left( x+1 \right)\left( x-9 \right)=0\]
Case 1:
\[x=-1\]
Argument of \[{{\log }_{9}}\left( x \right)\] is negative in this case which is not possible.
Case 2:
\[x=9\]
Arguments of both logs are positive in this case.
Hence, the solution of the equation is \[x=9\].
