Solve $LaTeX: \displaystyle \log_{6}(x + 237)+\log_{6}(x + 26) = 5$ .

Using logarithmic properties and expanding the argument gives $LaTeX: \displaystyle \log_{6}(x^{2} + 263 x + 6162)=5$ . Making both sides an exponent on the base gives $LaTeX: \displaystyle x^{2} + 263 x + 6162=6^{5}$ . Expanding and setting equal to zero gives $LaTeX: \displaystyle x^{2} + 263 x - 1614=0$ . Factoring gives $LaTeX: \displaystyle \left(x - 6\right) \left(x + 269\right)=0$ . Solving gives the two possible solutions $LaTeX: \displaystyle x = -269$ and $LaTeX: \displaystyle x = 6$ . The domain of the original is $LaTeX: \displaystyle \left(-237, \infty\right) \bigcap \left(-26, \infty\right)=\left(-26, \infty\right)$ . Checking if each possible solution is in the domain gives: $LaTeX: \displaystyle x = -269$ is not a solution. $LaTeX: \displaystyle x=6$ is a solution.