Solve the equation $LaTeX: \displaystyle \log_{7}(x + 2386)-\log_{7}(x + 34)=2$ .

Using the quotient property of logarithms gives $LaTeX: \displaystyle \log_{7}\frac{x + 2386}{x + 34} = 2$ . Making both sides of the equation exponents on the base $LaTeX: \displaystyle 7$ gives $LaTeX: \displaystyle \frac{x + 2386}{x + 34}=49$ . Clearing the fractions by multiplying by the LCD gives $LaTeX: \displaystyle x + 2386=49 x + 1666$ . Isolating $LaTeX: \displaystyle x$ gives $LaTeX: \displaystyle x = 15$ .