Solve $LaTeX: \displaystyle \log_{6}(x + 24)+\log_{6}(x + 235) = 5$ .

Using logarithmic properties and expanding the argument gives $LaTeX: \displaystyle \log_{6}(x^{2} + 259 x + 5640)=5$ . Making both sides an exponent on the base gives $LaTeX: \displaystyle x^{2} + 259 x + 5640=6^{5}$ . Expanding and setting equal to zero gives $LaTeX: \displaystyle x^{2} + 259 x - 2136=0$ . Factoring gives $LaTeX: \displaystyle \left(x - 8\right) \left(x + 267\right)=0$ . Solving gives the two possible solutions $LaTeX: \displaystyle x = -267$ and $LaTeX: \displaystyle x = 8$ . The domain of the original is $LaTeX: \displaystyle \left(-24, \infty\right) \bigcap \left(-235, \infty\right)=\left(-24, \infty\right)$ . Checking if each possible solution is in the domain gives: $LaTeX: \displaystyle x = -267$ is not a solution. $LaTeX: \displaystyle x=8$ is a solution.