Solve $LaTeX: \displaystyle \log_{20}(x + 622)+\log_{20}(x + 253) = 4$ .

Using logarithmic properties and expanding the argument gives $LaTeX: \displaystyle \log_{20}(x^{2} + 875 x + 157366)=4$ . Making both sides an exponent on the base gives $LaTeX: \displaystyle x^{2} + 875 x + 157366=20^{4}$ . Expanding and setting equal to zero gives $LaTeX: \displaystyle x^{2} + 875 x - 2634=0$ . Factoring gives $LaTeX: \displaystyle \left(x - 3\right) \left(x + 878\right)=0$ . Solving gives the two possible solutions $LaTeX: \displaystyle x = -878$ and $LaTeX: \displaystyle x = 3$ . The domain of the original is $LaTeX: \displaystyle \left(-622, \infty\right) \bigcap \left(-253, \infty\right)=\left(-253, \infty\right)$ . Checking if each possible solution is in the domain gives: $LaTeX: \displaystyle x = -878$ is not a solution. $LaTeX: \displaystyle x=3$ is a solution.