Solve the equation $LaTeX: \displaystyle \log_{5}(x + 3102)-\log_{5}(x + 102)=2$ .

Using the quotient property of logarithms gives $LaTeX: \displaystyle \log_{5}\frac{x + 3102}{x + 102} = 2$ . Making both sides of the equation exponents on the base $LaTeX: \displaystyle 5$ gives $LaTeX: \displaystyle \frac{x + 3102}{x + 102}=25$ . Clearing the fractions by multiplying by the LCD gives $LaTeX: \displaystyle x + 3102=25 x + 2550$ . Isolating $LaTeX: \displaystyle x$ gives $LaTeX: \displaystyle x = 23$ .