- Level: International Baccalaureate
- Subject: Maths
- Word count: 1141
In this investigation I will be examining logarithms and their bases. The purpose of examining logarithmic bases is to find patterns between the bases, its exponents and the product.
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Introduction
Mathematics SL Portfolio
Type I: Mathematical Investigation
Logarithm Bases
Mathematical Investigation: Logarithm Bases
In this investigation I will be examining logarithms and their bases. The purpose of examining logarithmic bases is to find patterns between the bases, its exponents and the product. Throughout this investigation I will be looking at different sets of logarithms and trying to determine a pattern. After I have found a pattern I will test its validity by applying the pattern to other sets of logarithms. First I’m going to look at four sequences of logarithms and then continue the pattern for two more terms.
- log28, log48, log88, log168, log328,log648, log1288
- log381, log981, log2781, log8181, log24381, log72981
- log525, log2525, log12525, log62525, log312525, log1562525
- logmmk, logm2mk, logm3mk, logm4mk, logm5mk, logm6mk
After studying the four sequences I noticed a similar pattern in each of them. In sequence 1 each term increased by multiplying 2 to the previous term’s base. But to express the sequence in
Middle
I will now calculate the values for other sequences and try to find a pattern on how to obtain the answers. All answers are written in the form , where p, q Z.
1. log464 log864 log3264
= 3 = 2 =
2. log749 log4949 log34349
= 2 = 1 =
3. log125 log125 log125
= - 3 = -1 = -
4. log8512 log2512 log16512
= 3 = 9 =
After analyzing the above sequences, I noticed a pattern between the first two answers and the third answer. To find the third answer it is necessary to multiply the first two answers. The value after multiplying the first two answers produces the numerator for the third answer. To find the denominator for the third answer it is necessary to add up the first two answers. Thus, the third answer is usually written in the form which in some cases can be simplified further. To prove this theory I will create two more sequences which follow the same pattern as the sequences above. Then I will evaluate the terms to prove my theory.
1. log6216 log36216 log216216
= 3 = 1.5 or = 1
2. log9729 log3
Conclusion
Throughout this portfolio I have investigated the patterns between logarithms and their bases. To find these patterns I looked at different sequences of logarithms. Then I found relationships between the values of the first two terms and how that value relates to the third term. By following this process I was able to find a general statement and test its validity. Then I described the limitations as well as the scope of the general statement. So, by completing this portfolio I was able to understand a lot about logarithms, their bases and the relationships between them.
Fig. 1.1
Fig. 1.2
Fig. 1.3
This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.
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