Solve $LaTeX: \displaystyle \log_{15}(x + 3117)+\log_{15}(x + 235) = 5$ .

Using logarithmic properties and expanding the argument gives $LaTeX: \displaystyle \log_{15}(x^{2} + 3352 x + 732495)=5$ . Making both sides an exponent on the base gives $LaTeX: \displaystyle x^{2} + 3352 x + 732495=15^{5}$ . Expanding and setting equal to zero gives $LaTeX: \displaystyle x^{2} + 3352 x - 26880=0$ . Factoring gives $LaTeX: \displaystyle \left(x - 8\right) \left(x + 3360\right)=0$ . Solving gives the two possible solutions $LaTeX: \displaystyle x = -3360$ and $LaTeX: \displaystyle x = 8$ . The domain of the original is $LaTeX: \displaystyle \left(-3117, \infty\right) \bigcap \left(-235, \infty\right)=\left(-235, \infty\right)$ . Checking if each possible solution is in the domain gives: $LaTeX: \displaystyle x = -3360$ is not a solution. $LaTeX: \displaystyle x=8$ is a solution.