Solve $LaTeX: \displaystyle \log_{15}(x + 622)+\log_{15}(x + 78) = 4$ .

Using logarithmic properties and expanding the argument gives $LaTeX: \displaystyle \log_{15}(x^{2} + 700 x + 48516)=4$ . Making both sides an exponent on the base gives $LaTeX: \displaystyle x^{2} + 700 x + 48516=15^{4}$ . Expanding and setting equal to zero gives $LaTeX: \displaystyle x^{2} + 700 x - 2109=0$ . Factoring gives $LaTeX: \displaystyle \left(x - 3\right) \left(x + 703\right)=0$ . Solving gives the two possible solutions $LaTeX: \displaystyle x = -703$ and $LaTeX: \displaystyle x = 3$ . The domain of the original is $LaTeX: \displaystyle \left(-622, \infty\right) \bigcap \left(-78, \infty\right)=\left(-78, \infty\right)$ . Checking if each possible solution is in the domain gives: $LaTeX: \displaystyle x = -703$ is not a solution. $LaTeX: \displaystyle x=3$ is a solution.