- Level: International Baccalaureate
- Subject: Maths
- Word count: 828
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:
,…
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:
,…
Next, the same logarithmic sequence will be evaluated but more in depth, to try and find a common pattern.
Note: a pattern is already starting to show.
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:
The following equation where 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:
To confirm that this pattern stays true, multiple other sequences will be used with the same formula: .
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
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