Solve $LaTeX: \displaystyle \log_{15}(x + 18)+\log_{15}(x + 2) = 2$ .

Using logarithmic properties and expanding the argument gives $LaTeX: \displaystyle \log_{15}(x^{2} + 20 x + 36)=2$ . Making both sides an exponent on the base gives $LaTeX: \displaystyle x^{2} + 20 x + 36=15^{2}$ . Expanding and setting equal to zero gives $LaTeX: \displaystyle x^{2} + 20 x - 189=0$ . Factoring gives $LaTeX: \displaystyle \left(x - 7\right) \left(x + 27\right)=0$ . Solving gives the two possible solutions $LaTeX: \displaystyle x = -27$ and $LaTeX: \displaystyle x = 7$ . The domain of the original is $LaTeX: \displaystyle \left(-18, \infty\right) \bigcap \left(-2, \infty\right)=\left(-2, \infty\right)$ . Checking if each possible solution is in the domain gives: $LaTeX: \displaystyle x = -27$ is not a solution. $LaTeX: \displaystyle x=7$ is a solution.