Solve $LaTeX: \displaystyle \log_{6}(x + 76)+\log_{6}(x + 11) = 4$ .

Using logarithmic properties and expanding the argument gives $LaTeX: \displaystyle \log_{6}(x^{2} + 87 x + 836)=4$ . Making both sides an exponent on the base gives $LaTeX: \displaystyle x^{2} + 87 x + 836=6^{4}$ . Expanding and setting equal to zero gives $LaTeX: \displaystyle x^{2} + 87 x - 460=0$ . Factoring gives $LaTeX: \displaystyle \left(x - 5\right) \left(x + 92\right)=0$ . Solving gives the two possible solutions $LaTeX: \displaystyle x = -92$ and $LaTeX: \displaystyle x = 5$ . The domain of the original is $LaTeX: \displaystyle \left(-76, \infty\right) \bigcap \left(-11, \infty\right)=\left(-11, \infty\right)$ . Checking if each possible solution is in the domain gives: $LaTeX: \displaystyle x = -92$ is not a solution. $LaTeX: \displaystyle x=5$ is a solution.