Solve $LaTeX: \displaystyle \log_{12}(x + 234)+\log_{12}(x + 1015) = 5$ .

Using logarithmic properties and expanding the argument gives $LaTeX: \displaystyle \log_{12}(x^{2} + 1249 x + 237510)=5$ . Making both sides an exponent on the base gives $LaTeX: \displaystyle x^{2} + 1249 x + 237510=12^{5}$ . Expanding and setting equal to zero gives $LaTeX: \displaystyle x^{2} + 1249 x - 11322=0$ . Factoring gives $LaTeX: \displaystyle \left(x - 9\right) \left(x + 1258\right)=0$ . Solving gives the two possible solutions $LaTeX: \displaystyle x = -1258$ and $LaTeX: \displaystyle x = 9$ . The domain of the original is $LaTeX: \displaystyle \left(-234, \infty\right) \bigcap \left(-1015, \infty\right)=\left(-234, \infty\right)$ . Checking if each possible solution is in the domain gives: $LaTeX: \displaystyle x = -1258$ is not a solution. $LaTeX: \displaystyle x=9$ is a solution.