Solve $LaTeX: \displaystyle \log_{12}(x + 241)+\log_{12}(x + 1022) = 5$ .

Using logarithmic properties and expanding the argument gives $LaTeX: \displaystyle \log_{12}(x^{2} + 1263 x + 246302)=5$ . Making both sides an exponent on the base gives $LaTeX: \displaystyle x^{2} + 1263 x + 246302=12^{5}$ . Expanding and setting equal to zero gives $LaTeX: \displaystyle x^{2} + 1263 x - 2530=0$ . Factoring gives $LaTeX: \displaystyle \left(x - 2\right) \left(x + 1265\right)=0$ . Solving gives the two possible solutions $LaTeX: \displaystyle x = -1265$ and $LaTeX: \displaystyle x = 2$ . The domain of the original is $LaTeX: \displaystyle \left(-241, \infty\right) \bigcap \left(-1022, \infty\right)=\left(-241, \infty\right)$ . Checking if each possible solution is in the domain gives: $LaTeX: \displaystyle x = -1265$ is not a solution. $LaTeX: \displaystyle x=2$ is a solution.