Solve the equation $LaTeX: \displaystyle \log_{6}(x + 7757)-\log_{6}(x + 197)=2$ .

Using the quotient property of logarithms gives $LaTeX: \displaystyle \log_{6}\frac{x + 7757}{x + 197} = 2$ . Making both sides of the equation exponents on the base $LaTeX: \displaystyle 6$ gives $LaTeX: \displaystyle \frac{x + 7757}{x + 197}=36$ . Clearing the fractions by multiplying by the LCD gives $LaTeX: \displaystyle x + 7757=36 x + 7092$ . Isolating $LaTeX: \displaystyle x$ gives $LaTeX: \displaystyle x = 19$ .