Solve $LaTeX: \displaystyle \log_{10}(x + 618)+\log_{10}(x + 9) = 4$ .

Using logarithmic properties and expanding the argument gives $LaTeX: \displaystyle \log_{10}(x^{2} + 627 x + 5562)=4$ . Making both sides an exponent on the base gives $LaTeX: \displaystyle x^{2} + 627 x + 5562=10^{4}$ . Expanding and setting equal to zero gives $LaTeX: \displaystyle x^{2} + 627 x - 4438=0$ . Factoring gives $LaTeX: \displaystyle \left(x - 7\right) \left(x + 634\right)=0$ . Solving gives the two possible solutions $LaTeX: \displaystyle x = -634$ and $LaTeX: \displaystyle x = 7$ . The domain of the original is $LaTeX: \displaystyle \left(-618, \infty\right) \bigcap \left(-9, \infty\right)=\left(-9, \infty\right)$ . Checking if each possible solution is in the domain gives: $LaTeX: \displaystyle x = -634$ is not a solution. $LaTeX: \displaystyle x=7$ is a solution.