Solve $LaTeX: \displaystyle \log_{20}(x + 116)+\log_{20}(x + 55) = 3$ .

Using logarithmic properties and expanding the argument gives $LaTeX: \displaystyle \log_{20}(x^{2} + 171 x + 6380)=3$ . Making both sides an exponent on the base gives $LaTeX: \displaystyle x^{2} + 171 x + 6380=20^{3}$ . Expanding and setting equal to zero gives $LaTeX: \displaystyle x^{2} + 171 x - 1620=0$ . Factoring gives $LaTeX: \displaystyle \left(x - 9\right) \left(x + 180\right)=0$ . Solving gives the two possible solutions $LaTeX: \displaystyle x = -180$ and $LaTeX: \displaystyle x = 9$ . The domain of the original is $LaTeX: \displaystyle \left(-116, \infty\right) \bigcap \left(-55, \infty\right)=\left(-55, \infty\right)$ . Checking if each possible solution is in the domain gives: $LaTeX: \displaystyle x = -180$ is not a solution. $LaTeX: \displaystyle x=9$ is a solution.