Solve $LaTeX: \displaystyle \log_{10}(x + 3118)+\log_{10}(x + 25) = 5$ .

Using logarithmic properties and expanding the argument gives $LaTeX: \displaystyle \log_{10}(x^{2} + 3143 x + 77950)=5$ . Making both sides an exponent on the base gives $LaTeX: \displaystyle x^{2} + 3143 x + 77950=10^{5}$ . Expanding and setting equal to zero gives $LaTeX: \displaystyle x^{2} + 3143 x - 22050=0$ . Factoring gives $LaTeX: \displaystyle \left(x - 7\right) \left(x + 3150\right)=0$ . Solving gives the two possible solutions $LaTeX: \displaystyle x = -3150$ and $LaTeX: \displaystyle x = 7$ . The domain of the original is $LaTeX: \displaystyle \left(-3118, \infty\right) \bigcap \left(-25, \infty\right)=\left(-25, \infty\right)$ . Checking if each possible solution is in the domain gives: $LaTeX: \displaystyle x = -3150$ is not a solution. $LaTeX: \displaystyle x=7$ is a solution.