Solve $LaTeX: \displaystyle \log_{6}(x + 18)+\log_{6}(x - 1) = 3$ .

Using logarithmic properties and expanding the argument gives $LaTeX: \displaystyle \log_{6}(x^{2} + 17 x - 18)=3$ . Making both sides an exponent on the base gives $LaTeX: \displaystyle x^{2} + 17 x - 18=6^{3}$ . Expanding and setting equal to zero gives $LaTeX: \displaystyle x^{2} + 17 x - 234=0$ . Factoring gives $LaTeX: \displaystyle \left(x - 9\right) \left(x + 26\right)=0$ . Solving gives the two possible solutions $LaTeX: \displaystyle x = -26$ and $LaTeX: \displaystyle x = 9$ . The domain of the original is $LaTeX: \displaystyle \left(-18, \infty\right) \bigcap \left(1, \infty\right)=\left(1, \infty\right)$ . Checking if each possible solution is in the domain gives: $LaTeX: \displaystyle x = -26$ is not a solution. $LaTeX: \displaystyle x=9$ is a solution.