Solve $LaTeX: \displaystyle \log_{20}(x + 23)+\log_{20}(x + 14) = 2$ .

Using logarithmic properties and expanding the argument gives $LaTeX: \displaystyle \log_{20}(x^{2} + 37 x + 322)=2$ . Making both sides an exponent on the base gives $LaTeX: \displaystyle x^{2} + 37 x + 322=20^{2}$ . Expanding and setting equal to zero gives $LaTeX: \displaystyle x^{2} + 37 x - 78=0$ . Factoring gives $LaTeX: \displaystyle \left(x - 2\right) \left(x + 39\right)=0$ . Solving gives the two possible solutions $LaTeX: \displaystyle x = -39$ and $LaTeX: \displaystyle x = 2$ . The domain of the original is $LaTeX: \displaystyle \left(-23, \infty\right) \bigcap \left(-14, \infty\right)=\left(-14, \infty\right)$ . Checking if each possible solution is in the domain gives: $LaTeX: \displaystyle x = -39$ is not a solution. $LaTeX: \displaystyle x=2$ is a solution.