Solve $LaTeX: \displaystyle \log_{ 10 }(x + 15) + \log_{ 10 }(x + 15) = 2$

Using the product rule for logarithms gives $LaTeX: \displaystyle \log_{ 10 }(\left(x + 15\right)^{2})$ and rewriting in exponential form gives $LaTeX: \displaystyle \left(x + 15\right)^{2} = 100$ expanding and setting the equation equal to zero gives $LaTeX: \displaystyle x^{2} + 30 x + 125 = 0$ . Factoring gives $LaTeX: \displaystyle \left(x + 5\right) \left(x + 25\right)=0$ . This gives two possible solutions $LaTeX: \displaystyle x=-25$ or $LaTeX: \displaystyle x=-5$ . $LaTeX: \displaystyle x=-25$ is an extraneous solution. The only soution is $LaTeX: \displaystyle x=-5$ .