Solve $LaTeX: \displaystyle \log_{15}(x + 615)+\log_{15}(x + 71) = 4$ .

Using logarithmic properties and expanding the argument gives $LaTeX: \displaystyle \log_{15}(x^{2} + 686 x + 43665)=4$ . Making both sides an exponent on the base gives $LaTeX: \displaystyle x^{2} + 686 x + 43665=15^{4}$ . Expanding and setting equal to zero gives $LaTeX: \displaystyle x^{2} + 686 x - 6960=0$ . Factoring gives $LaTeX: \displaystyle \left(x - 10\right) \left(x + 696\right)=0$ . Solving gives the two possible solutions $LaTeX: \displaystyle x = -696$ and $LaTeX: \displaystyle x = 10$ . The domain of the original is $LaTeX: \displaystyle \left(-615, \infty\right) \bigcap \left(-71, \infty\right)=\left(-71, \infty\right)$ . Checking if each possible solution is in the domain gives: $LaTeX: \displaystyle x = -696$ is not a solution. $LaTeX: \displaystyle x=10$ is a solution.