Solve $LaTeX: \displaystyle \log_{8}(x + 10)+\log_{8}(x + 250) = 4$ .

Using logarithmic properties and expanding the argument gives $LaTeX: \displaystyle \log_{8}(x^{2} + 260 x + 2500)=4$ . Making both sides an exponent on the base gives $LaTeX: \displaystyle x^{2} + 260 x + 2500=8^{4}$ . Expanding and setting equal to zero gives $LaTeX: \displaystyle x^{2} + 260 x - 1596=0$ . Factoring gives $LaTeX: \displaystyle \left(x - 6\right) \left(x + 266\right)=0$ . Solving gives the two possible solutions $LaTeX: \displaystyle x = -266$ and $LaTeX: \displaystyle x = 6$ . The domain of the original is $LaTeX: \displaystyle \left(-10, \infty\right) \bigcap \left(-250, \infty\right)=\left(-10, \infty\right)$ . Checking if each possible solution is in the domain gives: $LaTeX: \displaystyle x = -266$ is not a solution. $LaTeX: \displaystyle x=6$ is a solution.