Solve $LaTeX: \displaystyle \log_{10}(x + 115)+\log_{10}(x - 2) = 3$ .

Using logarithmic properties and expanding the argument gives $LaTeX: \displaystyle \log_{10}(x^{2} + 113 x - 230)=3$ . Making both sides an exponent on the base gives $LaTeX: \displaystyle x^{2} + 113 x - 230=10^{3}$ . Expanding and setting equal to zero gives $LaTeX: \displaystyle x^{2} + 113 x - 1230=0$ . Factoring gives $LaTeX: \displaystyle \left(x - 10\right) \left(x + 123\right)=0$ . Solving gives the two possible solutions $LaTeX: \displaystyle x = -123$ and $LaTeX: \displaystyle x = 10$ . The domain of the original is $LaTeX: \displaystyle \left(-115, \infty\right) \bigcap \left(2, \infty\right)=\left(2, \infty\right)$ . Checking if each possible solution is in the domain gives: $LaTeX: \displaystyle x = -123$ is not a solution. $LaTeX: \displaystyle x=10$ is a solution.