Solve $LaTeX: \displaystyle \log_{10}(x + 620)+\log_{10}(x + 11) = 4$ .

Using logarithmic properties and expanding the argument gives $LaTeX: \displaystyle \log_{10}(x^{2} + 631 x + 6820)=4$ . Making both sides an exponent on the base gives $LaTeX: \displaystyle x^{2} + 631 x + 6820=10^{4}$ . Expanding and setting equal to zero gives $LaTeX: \displaystyle x^{2} + 631 x - 3180=0$ . Factoring gives $LaTeX: \displaystyle \left(x - 5\right) \left(x + 636\right)=0$ . Solving gives the two possible solutions $LaTeX: \displaystyle x = -636$ and $LaTeX: \displaystyle x = 5$ . The domain of the original is $LaTeX: \displaystyle \left(-620, \infty\right) \bigcap \left(-11, \infty\right)=\left(-11, \infty\right)$ . Checking if each possible solution is in the domain gives: $LaTeX: \displaystyle x = -636$ is not a solution. $LaTeX: \displaystyle x=5$ is a solution.