Solve $LaTeX: \displaystyle \log_{8}(x + 1016)+\log_{8}(x + 24) = 5$ .

Using logarithmic properties and expanding the argument gives $LaTeX: \displaystyle \log_{8}(x^{2} + 1040 x + 24384)=5$ . Making both sides an exponent on the base gives $LaTeX: \displaystyle x^{2} + 1040 x + 24384=8^{5}$ . Expanding and setting equal to zero gives $LaTeX: \displaystyle x^{2} + 1040 x - 8384=0$ . Factoring gives $LaTeX: \displaystyle \left(x - 8\right) \left(x + 1048\right)=0$ . Solving gives the two possible solutions $LaTeX: \displaystyle x = -1048$ and $LaTeX: \displaystyle x = 8$ . The domain of the original is $LaTeX: \displaystyle \left(-1016, \infty\right) \bigcap \left(-24, \infty\right)=\left(-24, \infty\right)$ . Checking if each possible solution is in the domain gives: $LaTeX: \displaystyle x = -1048$ is not a solution. $LaTeX: \displaystyle x=8$ is a solution.