Solve $LaTeX: \displaystyle \log_{12}(x + 1020)+\log_{12}(x + 239) = 5$ .

Using logarithmic properties and expanding the argument gives $LaTeX: \displaystyle \log_{12}(x^{2} + 1259 x + 243780)=5$ . Making both sides an exponent on the base gives $LaTeX: \displaystyle x^{2} + 1259 x + 243780=12^{5}$ . Expanding and setting equal to zero gives $LaTeX: \displaystyle x^{2} + 1259 x - 5052=0$ . Factoring gives $LaTeX: \displaystyle \left(x - 4\right) \left(x + 1263\right)=0$ . Solving gives the two possible solutions $LaTeX: \displaystyle x = -1263$ and $LaTeX: \displaystyle x = 4$ . The domain of the original is $LaTeX: \displaystyle \left(-1020, \infty\right) \bigcap \left(-239, \infty\right)=\left(-239, \infty\right)$ . Checking if each possible solution is in the domain gives: $LaTeX: \displaystyle x = -1263$ is not a solution. $LaTeX: \displaystyle x=4$ is a solution.