Solve $LaTeX: \displaystyle \log_{8}(x + 54)+\log_{8}(x - 2) = 3$ .

Using logarithmic properties and expanding the argument gives $LaTeX: \displaystyle \log_{8}(x^{2} + 52 x - 108)=3$ . Making both sides an exponent on the base gives $LaTeX: \displaystyle x^{2} + 52 x - 108=8^{3}$ . Expanding and setting equal to zero gives $LaTeX: \displaystyle x^{2} + 52 x - 620=0$ . Factoring gives $LaTeX: \displaystyle \left(x - 10\right) \left(x + 62\right)=0$ . Solving gives the two possible solutions $LaTeX: \displaystyle x = -62$ and $LaTeX: \displaystyle x = 10$ . The domain of the original is $LaTeX: \displaystyle \left(-54, \infty\right) \bigcap \left(2, \infty\right)=\left(2, \infty\right)$ . Checking if each possible solution is in the domain gives: $LaTeX: \displaystyle x = -62$ is not a solution. $LaTeX: \displaystyle x=10$ is a solution.