Solve $LaTeX: \displaystyle \log_{10}(x + 616)+\log_{10}(x + 7) = 4$ .

Using logarithmic properties and expanding the argument gives $LaTeX: \displaystyle \log_{10}(x^{2} + 623 x + 4312)=4$ . Making both sides an exponent on the base gives $LaTeX: \displaystyle x^{2} + 623 x + 4312=10^{4}$ . Expanding and setting equal to zero gives $LaTeX: \displaystyle x^{2} + 623 x - 5688=0$ . Factoring gives $LaTeX: \displaystyle \left(x - 9\right) \left(x + 632\right)=0$ . Solving gives the two possible solutions $LaTeX: \displaystyle x = -632$ and $LaTeX: \displaystyle x = 9$ . The domain of the original is $LaTeX: \displaystyle \left(-616, \infty\right) \bigcap \left(-7, \infty\right)=\left(-7, \infty\right)$ . Checking if each possible solution is in the domain gives: $LaTeX: \displaystyle x = -632$ is not a solution. $LaTeX: \displaystyle x=9$ is a solution.