Solve $LaTeX: \displaystyle \log_{8}(x)+\log_{8}(x + 56) = 3$ .

Using logarithmic properties and expanding the argument gives $LaTeX: \displaystyle \log_{8}(x^{2} + 56 x)=3$ . Making both sides an exponent on the base gives $LaTeX: \displaystyle x^{2} + 56 x=8^{3}$ . Expanding and setting equal to zero gives $LaTeX: \displaystyle x^{2} + 56 x - 512=0$ . Factoring gives $LaTeX: \displaystyle \left(x - 8\right) \left(x + 64\right)=0$ . Solving gives the two possible solutions $LaTeX: \displaystyle x = -64$ and $LaTeX: \displaystyle x = 8$ . The domain of the original is $LaTeX: \displaystyle \left(0, \infty\right) \bigcap \left(-56, \infty\right)=\left(0, \infty\right)$ . Checking if each possible solution is in the domain gives: $LaTeX: \displaystyle x = -64$ is not a solution. $LaTeX: \displaystyle x=8$ is a solution.