Solve $LaTeX: \displaystyle \log_{12}(x + 253)+\log_{12}(x + 78) = 4$ .

Using logarithmic properties and expanding the argument gives $LaTeX: \displaystyle \log_{12}(x^{2} + 331 x + 19734)=4$ . Making both sides an exponent on the base gives $LaTeX: \displaystyle x^{2} + 331 x + 19734=12^{4}$ . Expanding and setting equal to zero gives $LaTeX: \displaystyle x^{2} + 331 x - 1002=0$ . Factoring gives $LaTeX: \displaystyle \left(x - 3\right) \left(x + 334\right)=0$ . Solving gives the two possible solutions $LaTeX: \displaystyle x = -334$ and $LaTeX: \displaystyle x = 3$ . The domain of the original is $LaTeX: \displaystyle \left(-253, \infty\right) \bigcap \left(-78, \infty\right)=\left(-78, \infty\right)$ . Checking if each possible solution is in the domain gives: $LaTeX: \displaystyle x = -334$ is not a solution. $LaTeX: \displaystyle x=3$ is a solution.