Solve $LaTeX: \displaystyle \log_{12}(x + 59)+\log_{12}(x + 22) = 3$ .

Using logarithmic properties and expanding the argument gives $LaTeX: \displaystyle \log_{12}(x^{2} + 81 x + 1298)=3$ . Making both sides an exponent on the base gives $LaTeX: \displaystyle x^{2} + 81 x + 1298=12^{3}$ . Expanding and setting equal to zero gives $LaTeX: \displaystyle x^{2} + 81 x - 430=0$ . Factoring gives $LaTeX: \displaystyle \left(x - 5\right) \left(x + 86\right)=0$ . Solving gives the two possible solutions $LaTeX: \displaystyle x = -86$ and $LaTeX: \displaystyle x = 5$ . The domain of the original is $LaTeX: \displaystyle \left(-59, \infty\right) \bigcap \left(-22, \infty\right)=\left(-22, \infty\right)$ . Checking if each possible solution is in the domain gives: $LaTeX: \displaystyle x = -86$ is not a solution. $LaTeX: \displaystyle x=5$ is a solution.