If one solution has 100 times as many hydrogen ions as another solution, what is the difference, in pH units between the two solution?

Use the fact that pH = log { 1 / [H+] }. Call x the concentration of hydrogen ions of one solution and 100x the concentration of hydrogen ions of the other solution. The pH of the solution with hydrogen concentration x is pH1 = log {1 / x}. The pH of the solution with hydrogen concentration 100x is pH2 = log {1 / 100x}. Now calculate pH2 - pH1 = log {1/x} - log {1 / 100x}. When you apply the logarithms properties you get pH2 - pH1 = log {1/x} - log {1/x} - log {1/100} = - (-2) = 2. So the answer is that if one solutions has 100 times as many hydrogen ions as another solution, the difference in pH units between the two solutions is 2.