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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.

  1. log28,  log48,  log88,  log168,  log328,log648,  log1288
  1. log381,  log981,  log2781,  log8181, log24381,  log72981
  1. log525,  log2525,  log12525,  log62525,  log312525,  log1562525
  1. 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

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Middle

 and its point of intersection also fit the theory mentioned above.

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 formimage00.png , where p, q image01.png Z.

1. log464                  log864                    log3264

   = 3                        = 2                          = image04.png

2. log749                  log4949                  log34349

    = 2                       = 1                         = image05.png

3. logimage06.png125               logimage07.png125             logimage08.png125

    = - 3                     = -1                        = -image10.png

4. log8512                log2512                log16512

    = 3                       = 9                        = image11.png

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 image00.png 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 image12.png            = 1

image13.png

2.  log9729              log3

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Conclusion

image28.png. This applies to the general statement because it is impossible to divide a number by 0. Thus, these are all the limitations on the general statement.

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

image29.png

Fig. 1.2

image30.png

Fig. 1.3

image31.png

...read more.

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