In this lesson, students make a conjecture about combining a difference of logarithms into a single logarithmic expression. First, students examine a pattern of logarithm differences that have integer values. Then, students write a conjecture about the pattern they notice and use the conjecture to evaluate logarithms that would be impossible for them to evaluate without the pattern. Finally, students prove their conjecture. They name the pattern the quotient rule, which states that the difference of logarithms with the same base is equivalent to a logarithm with the same base of the quotient of the arguments.

As students notice the pattern in logarithms, they are making use of repeated reasoning (MP8) to generalize the pattern into a conjecture.