Solve $LaTeX: \displaystyle \log_{10}(x + 617)+\log_{10}(x + 8) = 4$ .

Using logarithmic properties and expanding the argument gives $LaTeX: \displaystyle \log_{10}(x^{2} + 625 x + 4936)=4$ . Making both sides an exponent on the base gives $LaTeX: \displaystyle x^{2} + 625 x + 4936=10^{4}$ . Expanding and setting equal to zero gives $LaTeX: \displaystyle x^{2} + 625 x - 5064=0$ . Factoring gives $LaTeX: \displaystyle \left(x - 8\right) \left(x + 633\right)=0$ . Solving gives the two possible solutions $LaTeX: \displaystyle x = -633$ and $LaTeX: \displaystyle x = 8$ . The domain of the original is $LaTeX: \displaystyle \left(-617, \infty\right) \bigcap \left(-8, \infty\right)=\left(-8, \infty\right)$ . Checking if each possible solution is in the domain gives: $LaTeX: \displaystyle x = -633$ is not a solution. $LaTeX: \displaystyle x=8$ is a solution.