Solve the equation $LaTeX: \displaystyle \log_{5}(x + 113)-\log_{5}(x + 13)=1$ .

Using the quotient property of logarithms gives $LaTeX: \displaystyle \log_{5}\frac{x + 113}{x + 13} = 1$ . Making both sides of the equation exponents on the base $LaTeX: \displaystyle 5$ gives $LaTeX: \displaystyle \frac{x + 113}{x + 13}=5$ . Clearing the fractions by multiplying by the LCD gives $LaTeX: \displaystyle x + 113=5 x + 65$ . Isolating $LaTeX: \displaystyle x$ gives $LaTeX: \displaystyle x = 12$ .