Solve the equation $LaTeX: \displaystyle \log_{10}(x + 10001)-\log_{10}(x + 1001)=1$ .

Using the quotient property of logarithms gives $LaTeX: \displaystyle \log_{10}\frac{x + 10001}{x + 1001} = 1$ . Making both sides of the equation exponents on the base $LaTeX: \displaystyle 10$ gives $LaTeX: \displaystyle \frac{x + 10001}{x + 1001}=10$ . Clearing the fractions by multiplying by the LCD gives $LaTeX: \displaystyle x + 10001=10 x + 10010$ . Isolating $LaTeX: \displaystyle x$ gives $LaTeX: \displaystyle x = -1$ .