Solve $LaTeX: \displaystyle \log_{12}(x + 54)+\log_{12}(x + 17) = 3$ .

Using logarithmic properties and expanding the argument gives $LaTeX: \displaystyle \log_{12}(x^{2} + 71 x + 918)=3$ . Making both sides an exponent on the base gives $LaTeX: \displaystyle x^{2} + 71 x + 918=12^{3}$ . Expanding and setting equal to zero gives $LaTeX: \displaystyle x^{2} + 71 x - 810=0$ . Factoring gives $LaTeX: \displaystyle \left(x - 10\right) \left(x + 81\right)=0$ . Solving gives the two possible solutions $LaTeX: \displaystyle x = -81$ and $LaTeX: \displaystyle x = 10$ . The domain of the original is $LaTeX: \displaystyle \left(-54, \infty\right) \bigcap \left(-17, \infty\right)=\left(-17, \infty\right)$ . Checking if each possible solution is in the domain gives: $LaTeX: \displaystyle x = -81$ is not a solution. $LaTeX: \displaystyle x=10$ is a solution.