Logarithm Bases

        In the beginning of this problem, we are given multiple sequences of logarithms, and are told to write down the next two terms of each sequence. Here is a table of the first sequence, including the next two terms and the numerical equivalence of each term:

I created the last two terms in this sequence, terms 6 and 7, simply by doubling the base of the logarithm for each term. In more proper mathematical terms, I used the formula , where n is the number of the term. I noticed that each numerical equivalence seems to have 3 as the numerator, and the number of the term as the denominator. In this manner, I created a formula to find the numerical equivalence for the nth term of the sequence in the form , where both p and q are integers: .

        I used this method of investigation for the other two sequences as well. Here are tables for each of the two sequences:

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I noticed that in both of these sequences as well, the numerical equivalences seem to have a constant integer as the numerator, and the number of the term as the denominator. Therefore, the formula for the first sequence is , and the formula for the second sequence is .

        Now, we are asked to calculate a set of many logarithms, in the form , where p and q are both integers. The set is as follows:

Finding the numerical equivalences of the logarithms was very easy; I simply calculated them using a graphing calculator. The challenging part arose ...

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