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Logarithm Bases Math IA

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Introduction

Logarithmic Sequences IA

Andrew Cherny

Math SL

Scott Learned

1/26/11

In math class, I was given the assignment to evaluate multiple logarithmic sequences to see if any patterns were evident within these sequences.  Using logarithmic rules that I previously learned in math class, I was able to discover multiple patterns within each logarithmic expression.

To begin, I was given the general logarithmic expression:

image00.pngimage01.png,…

In order to establish patterns within this general logarithmic expression, I will use multiple examples to help establish a common pattern between all the examples. The first sequence is as followed:

image12.png,…

        Next, the same logarithmic sequence will be evaluated but more in depth, to try and find a common pattern.

image22.png

image32.png

image41.png

image43.png

image44.png

        Note: a pattern is already starting to show.

...read more.

Middle

Using the first sequence, I was able to discover that when you multiply the bases of the first and second log, you get the base of the third log, for example: 4 x 8 = 32.

To evaluate the first logarithmic sequence more in depth to try and find a pattern I did the following:

image23.png

image24.png

image25.png

        The following equation  image26.pngwhere c is the denominator of the answer on the first log, and where d is the denominator of the answer of the second log within the sequence. While x is the third term. Thus c = 3 and d= 2. To make more sense, here is the equation: image27.png

To confirm that this pattern stays true, multiple other sequences will be used with the same formula: image26.png.

...read more.

Conclusion

>

256

0.375

9

512

0.333333333

10

1024

0.3

11

2048

0.272727273

12

4096

0.25

13

8192

0.230769231

14

16384

0.214285714

15

32768

0.2

16

65536

0.1875

17

131072

0.176470588

18

262144

0.166666667

19

524288

0.157894737

20

1048576

0.15

21

2097152

0.142857143

22

4194304

0.136363636

23

8388608

0.130434783

24

16777216

0.125

25

33554432

0.12

26

67108864

0.115384615

27

134217728

0.111111111

28

268435456

0.107142857

29

536870912

0.103448276

30

1073741824

0.1

31

2147483648

0.096774194

32

4294967296

0.09375

33

8589934592

0.090909091

34

17179869184

0.088235294

35

34359738368

0.085714286

36

68719476736

0.083333333

37

1.37439E+11

0.081081081

38

2.74878E+11

0.078947368

39

5.49756E+11

0.076923077

40

1.09951E+12

0.075

41

2.19902E+12

0.073170732

42

4.39805E+12

0.071428571

43

8.79609E+12

0.069767442

44

1.75922E+13

0.068181818

45

3.51844E+13

0.066666667

46

7.03687E+13

0.065217391

47

1.40737E+14

0.063829787

48

2.81475E+14

0.0625

...read more.

This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.

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