Solve $LaTeX: \displaystyle \log_{12}(x + 249)+\log_{12}(x + 74) = 4$ .

Using logarithmic properties and expanding the argument gives $LaTeX: \displaystyle \log_{12}(x^{2} + 323 x + 18426)=4$ . Making both sides an exponent on the base gives $LaTeX: \displaystyle x^{2} + 323 x + 18426=12^{4}$ . Expanding and setting equal to zero gives $LaTeX: \displaystyle x^{2} + 323 x - 2310=0$ . Factoring gives $LaTeX: \displaystyle \left(x - 7\right) \left(x + 330\right)=0$ . Solving gives the two possible solutions $LaTeX: \displaystyle x = -330$ and $LaTeX: \displaystyle x = 7$ . The domain of the original is $LaTeX: \displaystyle \left(-249, \infty\right) \bigcap \left(-74, \infty\right)=\left(-74, \infty\right)$ . Checking if each possible solution is in the domain gives: $LaTeX: \displaystyle x = -330$ is not a solution. $LaTeX: \displaystyle x=7$ is a solution.