Solve $LaTeX: \displaystyle \log_{12}(x + 1017)+\log_{12}(x + 236) = 5$ .

Using logarithmic properties and expanding the argument gives $LaTeX: \displaystyle \log_{12}(x^{2} + 1253 x + 240012)=5$ . Making both sides an exponent on the base gives $LaTeX: \displaystyle x^{2} + 1253 x + 240012=12^{5}$ . Expanding and setting equal to zero gives $LaTeX: \displaystyle x^{2} + 1253 x - 8820=0$ . Factoring gives $LaTeX: \displaystyle \left(x - 7\right) \left(x + 1260\right)=0$ . Solving gives the two possible solutions $LaTeX: \displaystyle x = -1260$ and $LaTeX: \displaystyle x = 7$ . The domain of the original is $LaTeX: \displaystyle \left(-1017, \infty\right) \bigcap \left(-236, \infty\right)=\left(-236, \infty\right)$ . Checking if each possible solution is in the domain gives: $LaTeX: \displaystyle x = -1260$ is not a solution. $LaTeX: \displaystyle x=7$ is a solution.