Solve $LaTeX: \displaystyle \log_{12}(x + 251)+\log_{12}(x + 76) = 4$ .

Using logarithmic properties and expanding the argument gives $LaTeX: \displaystyle \log_{12}(x^{2} + 327 x + 19076)=4$ . Making both sides an exponent on the base gives $LaTeX: \displaystyle x^{2} + 327 x + 19076=12^{4}$ . Expanding and setting equal to zero gives $LaTeX: \displaystyle x^{2} + 327 x - 1660=0$ . Factoring gives $LaTeX: \displaystyle \left(x - 5\right) \left(x + 332\right)=0$ . Solving gives the two possible solutions $LaTeX: \displaystyle x = -332$ and $LaTeX: \displaystyle x = 5$ . The domain of the original is $LaTeX: \displaystyle \left(-251, \infty\right) \bigcap \left(-76, \infty\right)=\left(-76, \infty\right)$ . Checking if each possible solution is in the domain gives: $LaTeX: \displaystyle x = -332$ is not a solution. $LaTeX: \displaystyle x=5$ is a solution.