Solve $LaTeX: \displaystyle \log_{20}(x + 617)+\log_{20}(x + 248) = 4$ .

Using logarithmic properties and expanding the argument gives $LaTeX: \displaystyle \log_{20}(x^{2} + 865 x + 153016)=4$ . Making both sides an exponent on the base gives $LaTeX: \displaystyle x^{2} + 865 x + 153016=20^{4}$ . Expanding and setting equal to zero gives $LaTeX: \displaystyle x^{2} + 865 x - 6984=0$ . Factoring gives $LaTeX: \displaystyle \left(x - 8\right) \left(x + 873\right)=0$ . Solving gives the two possible solutions $LaTeX: \displaystyle x = -873$ and $LaTeX: \displaystyle x = 8$ . The domain of the original is $LaTeX: \displaystyle \left(-617, \infty\right) \bigcap \left(-248, \infty\right)=\left(-248, \infty\right)$ . Checking if each possible solution is in the domain gives: $LaTeX: \displaystyle x = -873$ is not a solution. $LaTeX: \displaystyle x=8$ is a solution.