Solve $LaTeX: \displaystyle \log_{10}(x + 3122)+\log_{10}(x + 29) = 5$ .

Using logarithmic properties and expanding the argument gives $LaTeX: \displaystyle \log_{10}(x^{2} + 3151 x + 90538)=5$ . Making both sides an exponent on the base gives $LaTeX: \displaystyle x^{2} + 3151 x + 90538=10^{5}$ . Expanding and setting equal to zero gives $LaTeX: \displaystyle x^{2} + 3151 x - 9462=0$ . Factoring gives $LaTeX: \displaystyle \left(x - 3\right) \left(x + 3154\right)=0$ . Solving gives the two possible solutions $LaTeX: \displaystyle x = -3154$ and $LaTeX: \displaystyle x = 3$ . The domain of the original is $LaTeX: \displaystyle \left(-3122, \infty\right) \bigcap \left(-29, \infty\right)=\left(-29, \infty\right)$ . Checking if each possible solution is in the domain gives: $LaTeX: \displaystyle x = -3154$ is not a solution. $LaTeX: \displaystyle x=3$ is a solution.