Solve $LaTeX: \displaystyle \log_{20}(x + 3122)+\log_{20}(x + 1021) = 5$ .

Using logarithmic properties and expanding the argument gives $LaTeX: \displaystyle \log_{20}(x^{2} + 4143 x + 3187562)=5$ . Making both sides an exponent on the base gives $LaTeX: \displaystyle x^{2} + 4143 x + 3187562=20^{5}$ . Expanding and setting equal to zero gives $LaTeX: \displaystyle x^{2} + 4143 x - 12438=0$ . Factoring gives $LaTeX: \displaystyle \left(x - 3\right) \left(x + 4146\right)=0$ . Solving gives the two possible solutions $LaTeX: \displaystyle x = -4146$ and $LaTeX: \displaystyle x = 3$ . The domain of the original is $LaTeX: \displaystyle \left(-3122, \infty\right) \bigcap \left(-1021, \infty\right)=\left(-1021, \infty\right)$ . Checking if each possible solution is in the domain gives: $LaTeX: \displaystyle x = -4146$ is not a solution. $LaTeX: \displaystyle x=3$ is a solution.