Solve $LaTeX: \displaystyle \log_{12}(x + 1018)+\log_{12}(x + 237) = 5$ .

Using logarithmic properties and expanding the argument gives $LaTeX: \displaystyle \log_{12}(x^{2} + 1255 x + 241266)=5$ . Making both sides an exponent on the base gives $LaTeX: \displaystyle x^{2} + 1255 x + 241266=12^{5}$ . Expanding and setting equal to zero gives $LaTeX: \displaystyle x^{2} + 1255 x - 7566=0$ . Factoring gives $LaTeX: \displaystyle \left(x - 6\right) \left(x + 1261\right)=0$ . Solving gives the two possible solutions $LaTeX: \displaystyle x = -1261$ and $LaTeX: \displaystyle x = 6$ . The domain of the original is $LaTeX: \displaystyle \left(-1018, \infty\right) \bigcap \left(-237, \infty\right)=\left(-237, \infty\right)$ . Checking if each possible solution is in the domain gives: $LaTeX: \displaystyle x = -1261$ is not a solution. $LaTeX: \displaystyle x=6$ is a solution.