Solve $LaTeX: \displaystyle \log_{ 5 }(x + 10) + \log_{ 5 }(x + 10) = 2$

Using the product rule for logarithms gives $LaTeX: \displaystyle \log_{ 5 }(\left(x + 10\right)^{2})$ and rewriting in exponential form gives $LaTeX: \displaystyle \left(x + 10\right)^{2} = 25$ expanding and setting the equation equal to zero gives $LaTeX: \displaystyle x^{2} + 20 x + 75 = 0$ . Factoring gives $LaTeX: \displaystyle \left(x + 5\right) \left(x + 15\right)=0$ . This gives two possible solutions $LaTeX: \displaystyle x=-15$ or $LaTeX: \displaystyle x=-5$ . $LaTeX: \displaystyle x=-15$ is an extraneous solution. The only soution is $LaTeX: \displaystyle x=-5$ .