Solve the equation $LaTeX: \displaystyle \log_{9}(x + 6564)-\log_{9}(x + 84)=2$ .

Using the quotient property of logarithms gives $LaTeX: \displaystyle \log_{9}\frac{x + 6564}{x + 84} = 2$ . Making both sides of the equation exponents on the base $LaTeX: \displaystyle 9$ gives $LaTeX: \displaystyle \frac{x + 6564}{x + 84}=81$ . Clearing the fractions by multiplying by the LCD gives $LaTeX: \displaystyle x + 6564=81 x + 6804$ . Isolating $LaTeX: \displaystyle x$ gives $LaTeX: \displaystyle x = -3$ .