Solve $LaTeX: \displaystyle \log_{6}(x - 1)+\log_{6}(x - 6) = 2$ .

Using logarithmic properties and expanding the argument gives $LaTeX: \displaystyle \log_{6}(x^{2} - 7 x + 6)=2$ . Making both sides an exponent on the base gives $LaTeX: \displaystyle x^{2} - 7 x + 6=6^{2}$ . Expanding and setting equal to zero gives $LaTeX: \displaystyle x^{2} - 7 x - 30=0$ . Factoring gives $LaTeX: \displaystyle \left(x - 10\right) \left(x + 3\right)=0$ . Solving gives the two possible solutions $LaTeX: \displaystyle x = -3$ and $LaTeX: \displaystyle x = 10$ . The domain of the original is $LaTeX: \displaystyle \left(1, \infty\right) \bigcap \left(6, \infty\right)=\left(6, \infty\right)$ . Checking if each possible solution is in the domain gives: $LaTeX: \displaystyle x = -3$ is not a solution. $LaTeX: \displaystyle x=10$ is a solution.