Solve the equation $LaTeX: \displaystyle \log_{10}(x + 100022)-\log_{10}(x + 1022)=2$ .

Using the quotient property of logarithms gives $LaTeX: \displaystyle \log_{10}\frac{x + 100022}{x + 1022} = 2$ . Making both sides of the equation exponents on the base $LaTeX: \displaystyle 10$ gives $LaTeX: \displaystyle \frac{x + 100022}{x + 1022}=100$ . Clearing the fractions by multiplying by the LCD gives $LaTeX: \displaystyle x + 100022=100 x + 102200$ . Isolating $LaTeX: \displaystyle x$ gives $LaTeX: \displaystyle x = -22$ .