Solve $LaTeX: \displaystyle \log_{ 12 }(x + 15) + \log_{ 12 }(x + 1731) = 4$

Using the product rule for logarithms gives $LaTeX: \displaystyle \log_{ 12 }(\left(x + 15\right) \left(x + 1731\right))$ and rewriting in exponential form gives $LaTeX: \displaystyle \left(x + 15\right) \left(x + 1731\right) = 20736$ expanding and setting the equation equal to zero gives $LaTeX: \displaystyle x^{2} + 1746 x + 5229 = 0$ . Factoring gives $LaTeX: \displaystyle \left(x + 3\right) \left(x + 1743\right)=0$ . This gives two possible solutions $LaTeX: \displaystyle x=-1743$ or $LaTeX: \displaystyle x=-3$ . $LaTeX: \displaystyle x=-1743$ is an extraneous solution. The only soution is $LaTeX: \displaystyle x=-3$ .