Solve $LaTeX: \displaystyle \log_{ 8 }(x + 10) + \log_{ 8 }(x + 514) = 4$

Using the product rule for logarithms gives $LaTeX: \displaystyle \log_{ 8 }(\left(x + 10\right) \left(x + 514\right))$ and rewriting in exponential form gives $LaTeX: \displaystyle \left(x + 10\right) \left(x + 514\right) = 4096$ expanding and setting the equation equal to zero gives $LaTeX: \displaystyle x^{2} + 524 x + 1044 = 0$ . Factoring gives $LaTeX: \displaystyle \left(x + 2\right) \left(x + 522\right)=0$ . This gives two possible solutions $LaTeX: \displaystyle x=-522$ or $LaTeX: \displaystyle x=-2$ . $LaTeX: \displaystyle x=-522$ is an extraneous solution. The only soution is $LaTeX: \displaystyle x=-2$ .