Solve $LaTeX: \displaystyle \log_{12}(x + 71)+\log_{12}(x + 246) = 4$ .

Using logarithmic properties and expanding the argument gives $LaTeX: \displaystyle \log_{12}(x^{2} + 317 x + 17466)=4$ . Making both sides an exponent on the base gives $LaTeX: \displaystyle x^{2} + 317 x + 17466=12^{4}$ . Expanding and setting equal to zero gives $LaTeX: \displaystyle x^{2} + 317 x - 3270=0$ . Factoring gives $LaTeX: \displaystyle \left(x - 10\right) \left(x + 327\right)=0$ . Solving gives the two possible solutions $LaTeX: \displaystyle x = -327$ and $LaTeX: \displaystyle x = 10$ . The domain of the original is $LaTeX: \displaystyle \left(-71, \infty\right) \bigcap \left(-246, \infty\right)=\left(-71, \infty\right)$ . Checking if each possible solution is in the domain gives: $LaTeX: \displaystyle x = -327$ is not a solution. $LaTeX: \displaystyle x=10$ is a solution.