Answer
The solution is $x=9$
Work Step by Step
$\log_{3}(x-8)+\log_{3}x=2$
Combine the sum on the left side as the logarithm of a product:
$\log_{3}x(x-8)=2$
Evaluate the product:
$\log_{3}(x^{2}-8x)=2$
Rewrite the equation in exponential form:
$x^{2}-8x=3^{2}$
$x^{2}-8x=9$
Take the $9$ to the left side:
$x^{2}-8x-9=0$
Solve by factoring:
$(x-9)(x+1)=0$
Set both factors equal to $0$ and solve both individual equations for $x$:
$x-9=0$
$x=9$
$x+1=0$
$x=-1$
The solutions found are $x=9$ and $x=-1$. Check these solutions by plugging them into the original equation.
$x=9$
$\log_{3}(9-8)+\log_{3}9=2$
$\log_{3}1+\log_{3}9=2$
$0+2=2$
$2=2$ True
$x=-1$
$\log_{3}(-1-8)+\log_{3}-1=2$ False
The solution is $x=9$
