Solve $LaTeX: \displaystyle \log_{8}(x + 6)+\log_{8}(x + 62) = 3$ .

Using logarithmic properties and expanding the argument gives $LaTeX: \displaystyle \log_{8}(x^{2} + 68 x + 372)=3$ . Making both sides an exponent on the base gives $LaTeX: \displaystyle x^{2} + 68 x + 372=8^{3}$ . Expanding and setting equal to zero gives $LaTeX: \displaystyle x^{2} + 68 x - 140=0$ . Factoring gives $LaTeX: \displaystyle \left(x - 2\right) \left(x + 70\right)=0$ . Solving gives the two possible solutions $LaTeX: \displaystyle x = -70$ and $LaTeX: \displaystyle x = 2$ . The domain of the original is $LaTeX: \displaystyle \left(-6, \infty\right) \bigcap \left(-62, \infty\right)=\left(-6, \infty\right)$ . Checking if each possible solution is in the domain gives: $LaTeX: \displaystyle x = -70$ is not a solution. $LaTeX: \displaystyle x=2$ is a solution.