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Logarithm Bases

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Introduction

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:

# of Term

1

2

3

4

5

6

7

Term

image00.png

image01.png

image19.png

image30.png

image41.png

image48.png

image57.png

Numerical Equivalence

image52.png

image71.png

image80.png

image02.png

image11.png

image14.png

image15.png

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 image16.pngimage16.png, 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 image17.pngimage17.png

...read more.

Middle

image38.png

Numerical Equivalence

image39.png

image40.png

image42.png

image43.png

image44.png

image45.png

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 image46.pngimage46.png, and the formula for the second sequence is image47.pngimage47.png.

        Now, we are asked to calculate a set of many logarithms, in the form image17.pngimage17.png, where p and q are both integers. The set is as follows:

Logarithm

image49.png

image50.png

image51.png

Numerical Equivalence

image52.png

image39.png

image53.png

Logarithm

image54.png

image55.png

image56.png

Numerical Equivalence

image39.png

image58.png

image42.png

Logarithm

image59.png

image60.png

image61.png

Numerical Equivalence

image62.png

image63.png

image64.png

Logarithm

image65.png

image66.png

image67.png

Numerical Equivalence

image52.png

image68.png

image69.png

Finding the numerical equivalences of the logarithms was very easy; I simply calculated them using a graphing calculator. The challenging part arose when I was asked to find a way to obtain the third answer in each row from the first two answers in each row. I was also told to let image10.pngimage10.png and image12.pngimage12.png, and to find a general statement that expresses image70.pngimage70.png, in terms of c and d.

...read more.

Conclusion

Logarithm

image33.png

image83.png

image84.png

Numerical Equivalence

2

-2

ERROR

This didn’t work, because c+d=0, and the general formula ended up with a 0 on the bottom. When I tried to solve the third logarithm using my calculator, it gave me an error message saying “Divide by 0”.  Therefore, image85.pngimage85.png and image03.pngimage03.png.

        One more thing that I thought of was that x must be greater than 0, because of the definition of a logarithm. There is no possible way to get any number to equal 0 or any negative number using only exponents. Therefore, image04.pngimage04.png.

        I couldn’t think of any more limitations of the general statement. Here are the limitations that I came up with:

image05.png

image06.png

image07.png

image08.png

image09.png

I will now repeat my general statement. If we let image10.pngimage10.png and image12.pngimage12.png, then

image13.pngimage13.png.

This statement only works with the limitations stated previously. It took me a lot of staring at the numbers and trying different values for a, b, and x, but eventually I discerned a pattern and arrived at my general statement.

...read more.

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