Solve $LaTeX: \displaystyle \log_{8}(x + 251)+\log_{8}(x + 11) = 4$ .

Using logarithmic properties and expanding the argument gives $LaTeX: \displaystyle \log_{8}(x^{2} + 262 x + 2761)=4$ . Making both sides an exponent on the base gives $LaTeX: \displaystyle x^{2} + 262 x + 2761=8^{4}$ . Expanding and setting equal to zero gives $LaTeX: \displaystyle x^{2} + 262 x - 1335=0$ . Factoring gives $LaTeX: \displaystyle \left(x - 5\right) \left(x + 267\right)=0$ . Solving gives the two possible solutions $LaTeX: \displaystyle x = -267$ and $LaTeX: \displaystyle x = 5$ . The domain of the original is $LaTeX: \displaystyle \left(-251, \infty\right) \bigcap \left(-11, \infty\right)=\left(-11, \infty\right)$ . Checking if each possible solution is in the domain gives: $LaTeX: \displaystyle x = -267$ is not a solution. $LaTeX: \displaystyle x=5$ is a solution.