Solve $LaTeX: \displaystyle \log_{12}(x + 7)+\log_{12}(x + 14) = 2$ .

Using logarithmic properties and expanding the argument gives $LaTeX: \displaystyle \log_{12}(x^{2} + 21 x + 98)=2$ . Making both sides an exponent on the base gives $LaTeX: \displaystyle x^{2} + 21 x + 98=12^{2}$ . Expanding and setting equal to zero gives $LaTeX: \displaystyle x^{2} + 21 x - 46=0$ . Factoring gives $LaTeX: \displaystyle \left(x - 2\right) \left(x + 23\right)=0$ . Solving gives the two possible solutions $LaTeX: \displaystyle x = -23$ and $LaTeX: \displaystyle x = 2$ . The domain of the original is $LaTeX: \displaystyle \left(-7, \infty\right) \bigcap \left(-14, \infty\right)=\left(-7, \infty\right)$ . Checking if each possible solution is in the domain gives: $LaTeX: \displaystyle x = -23$ is not a solution. $LaTeX: \displaystyle x=2$ is a solution.