Solve $LaTeX: \displaystyle \log_{12}(x + 1014)+\log_{12}(x + 233) = 5$ .

Using logarithmic properties and expanding the argument gives $LaTeX: \displaystyle \log_{12}(x^{2} + 1247 x + 236262)=5$ . Making both sides an exponent on the base gives $LaTeX: \displaystyle x^{2} + 1247 x + 236262=12^{5}$ . Expanding and setting equal to zero gives $LaTeX: \displaystyle x^{2} + 1247 x - 12570=0$ . Factoring gives $LaTeX: \displaystyle \left(x - 10\right) \left(x + 1257\right)=0$ . Solving gives the two possible solutions $LaTeX: \displaystyle x = -1257$ and $LaTeX: \displaystyle x = 10$ . The domain of the original is $LaTeX: \displaystyle \left(-1014, \infty\right) \bigcap \left(-233, \infty\right)=\left(-233, \infty\right)$ . Checking if each possible solution is in the domain gives: $LaTeX: \displaystyle x = -1257$ is not a solution. $LaTeX: \displaystyle x=10$ is a solution.