Solve $LaTeX: \displaystyle \log_{6}(x + 12)+\log_{6}(x + 77) = 4$ .

Using logarithmic properties and expanding the argument gives $LaTeX: \displaystyle \log_{6}(x^{2} + 89 x + 924)=4$ . Making both sides an exponent on the base gives $LaTeX: \displaystyle x^{2} + 89 x + 924=6^{4}$ . Expanding and setting equal to zero gives $LaTeX: \displaystyle x^{2} + 89 x - 372=0$ . Factoring gives $LaTeX: \displaystyle \left(x - 4\right) \left(x + 93\right)=0$ . Solving gives the two possible solutions $LaTeX: \displaystyle x = -93$ and $LaTeX: \displaystyle x = 4$ . The domain of the original is $LaTeX: \displaystyle \left(-12, \infty\right) \bigcap \left(-77, \infty\right)=\left(-12, \infty\right)$ . Checking if each possible solution is in the domain gives: $LaTeX: \displaystyle x = -93$ is not a solution. $LaTeX: \displaystyle x=4$ is a solution.