Solve $LaTeX: \displaystyle \log_{12}(x + 20)+\log_{12}(x + 57) = 3$ .

Using logarithmic properties and expanding the argument gives $LaTeX: \displaystyle \log_{12}(x^{2} + 77 x + 1140)=3$ . Making both sides an exponent on the base gives $LaTeX: \displaystyle x^{2} + 77 x + 1140=12^{3}$ . Expanding and setting equal to zero gives $LaTeX: \displaystyle x^{2} + 77 x - 588=0$ . Factoring gives $LaTeX: \displaystyle \left(x - 7\right) \left(x + 84\right)=0$ . Solving gives the two possible solutions $LaTeX: \displaystyle x = -84$ and $LaTeX: \displaystyle x = 7$ . The domain of the original is $LaTeX: \displaystyle \left(-20, \infty\right) \bigcap \left(-57, \infty\right)=\left(-20, \infty\right)$ . Checking if each possible solution is in the domain gives: $LaTeX: \displaystyle x = -84$ is not a solution. $LaTeX: \displaystyle x=7$ is a solution.