Solve $LaTeX: \displaystyle \log_{20}(x + 618)+\log_{20}(x + 249) = 4$ .

Using logarithmic properties and expanding the argument gives $LaTeX: \displaystyle \log_{20}(x^{2} + 867 x + 153882)=4$ . Making both sides an exponent on the base gives $LaTeX: \displaystyle x^{2} + 867 x + 153882=20^{4}$ . Expanding and setting equal to zero gives $LaTeX: \displaystyle x^{2} + 867 x - 6118=0$ . Factoring gives $LaTeX: \displaystyle \left(x - 7\right) \left(x + 874\right)=0$ . Solving gives the two possible solutions $LaTeX: \displaystyle x = -874$ and $LaTeX: \displaystyle x = 7$ . The domain of the original is $LaTeX: \displaystyle \left(-618, \infty\right) \bigcap \left(-249, \infty\right)=\left(-249, \infty\right)$ . Checking if each possible solution is in the domain gives: $LaTeX: \displaystyle x = -874$ is not a solution. $LaTeX: \displaystyle x=7$ is a solution.