Solve the equation $LaTeX: \displaystyle \log_{8}(x + 4078)-\log_{8}(x + 46)=2$ .

Using the quotient property of logarithms gives $LaTeX: \displaystyle \log_{8}\frac{x + 4078}{x + 46} = 2$ . Making both sides of the equation exponents on the base $LaTeX: \displaystyle 8$ gives $LaTeX: \displaystyle \frac{x + 4078}{x + 46}=64$ . Clearing the fractions by multiplying by the LCD gives $LaTeX: \displaystyle x + 4078=64 x + 2944$ . Isolating $LaTeX: \displaystyle x$ gives $LaTeX: \displaystyle x = 18$ .