Answer
$x=6$
Work Step by Step
$\log_{9}(x-5)+\log_{9}(x+3)=1$
Combine the logarithms on the left side of the equation as a product:
$\log_{9}(x-5)(x+3)=1$
$\log_{9}(x^{2}-2x-15)=1$
Write this equation in exponential form:
$x^{2}-2x-15=9^{1}$
$x^{2}-2x-15=9$
Take the $9$ to the left side of the equation:
$x^{2}-2x-15-9=0$
$x^{2}-2x-24=0$
Solve by factoring:
$(x+4)(x-6)=0$
We get two solutions:
$x=-4$ and $x=6$
We see that the initial equation is undefined when $x=-4$, so we can discard that solution.
Our final answer is $x=6$.