Solve $LaTeX: \displaystyle \log_{12}(x + 79)+\log_{12}(x + 254) = 4$ .

Using logarithmic properties and expanding the argument gives $LaTeX: \displaystyle \log_{12}(x^{2} + 333 x + 20066)=4$ . Making both sides an exponent on the base gives $LaTeX: \displaystyle x^{2} + 333 x + 20066=12^{4}$ . Expanding and setting equal to zero gives $LaTeX: \displaystyle x^{2} + 333 x - 670=0$ . Factoring gives $LaTeX: \displaystyle \left(x - 2\right) \left(x + 335\right)=0$ . Solving gives the two possible solutions $LaTeX: \displaystyle x = -335$ and $LaTeX: \displaystyle x = 2$ . The domain of the original is $LaTeX: \displaystyle \left(-79, \infty\right) \bigcap \left(-254, \infty\right)=\left(-79, \infty\right)$ . Checking if each possible solution is in the domain gives: $LaTeX: \displaystyle x = -335$ is not a solution. $LaTeX: \displaystyle x=2$ is a solution.