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Math Portfolio

Extracts from this document...

Introduction

Markowitz                                                                                                 0000602-013

                                   IB Math Standard Level Portfolio Assignment

                                           Type 1- Mathematical Investigation

                                                            Logarithm Bases

                                                           Jaclyn Markowitz

                                                IB Candidate ID: 0000602-013

I have completed this assignment in accordance with the Newark Academy honor code.

X___________________________.

This investigation will determine the relationship between different sets of sequences including logarithms. This investigation will be tested using technology and general mathematic relationships.

There are a few rules, which govern all the concepts of logarithms:

image00.png,  image01.png   where a>0, a≠1, b>0

image52.png

The sets of sequences are as follows:

log₂8,  log₄8,  log₈8,  log₁₆8,  log₃₂8

log₃81,  log₉81,  log₂₇81,  log₈₁81

log₅25,  log₂₅25,  log₁₂₅25,  log₆₂₅25

:

:

:

image92.png

By following these sequences a pattern can be shown. The base of each term in the sequences changes, but the number in the logarithm remains constant. The following two terms of each sequence were determined:

log₂8,  log₄8,  log₈8,  log₁₆8,  log₃₂8,  log₆₄8,  log₁₂₈8

log₃81,  log₉81,  log₂₇81,  log₈₁81,  log₂₄₃81,  log₇₂₉81

log₅25,  log₂₅25,  log₁₂₅25,  log₆₂₅25, image102.png,  log₁₅₆₂₅25

:

:

:

image111.png

Find an expression for the nth term of each sequence. Write down your expression in the form image120.png , where p, q image127.png. Justify your answers using technology.

Begin with the first sequence (log₂8,  log₄8,  log₈8,  log₁₆8,  log₃₂8,  log₆₄8,  log₁₂₈8)

...read more.

Middle

25

125

625

3125

15625

78125

3. image112.png

Use the change of base formula:

image113.png

Then apply the rule: image63.png

image114.png

image115.png cancels out on both sides


What remains:
image116.png

Here are two graphs to test its validity and accurateness:

Y=image117.png

image118.png

Y=image113.png

image119.png

Both graphs are identical towards each other, which indicates that both functions of the graph are the same.

4. Sequence expressed using the variables m, n, and k.

image04.png

Use  image52.png to change the base of the logarithm to ten.

image121.png

image122.pngcancels out.

What remains isimage123.png.  Derived from this, it can be concluded that the general expression for the nth term of each sequence in the form image124.png is image123.png.  

Justification of this statement using technology:

image125.png        (image04.png)

image126.png

image128.png               (image04.png)

image129.png

image130.png                 (image04.png)

image131.png

Now calculate the following, giving your answer in the form image120.png , where p, q, image132.png.

The answer was image133.png. This form will be used.

image134.pngimage04.png

image136.png

image137.pngimage04.png

image138.png


image139.pngimage04.png

image140.png


2.  image141.pngimage04.png

image143.png

image144.pngimage04.png


image145.png

image146.pngimage04.png

image147.png

3.  image130.pngimage04.png

image148.png

image03.pngimage04.png

image05.png

image06.pngimage04.png

image07.png


4. image08.pngimage04.png

image09.png

image10.pngimage04.png


image12.png

image13.pngimage04.png


image14.png

Describe how to obtain the third answer in each row from the first two answers. Create two more examples that fit the pattern above.

1.

image15.png

By recognizing that image16.png is the base in each of these logarithm, it is apparent that n=1 in the first logarithm and n=2 in the second logarithm.

...read more.

Conclusion

image76.png= error/ not possible

The example applies for a, that b≠1.

Example 3: a=4, b=8, c=-8.

image77.png    and   image78.png

image79.png

As seen in example 1, it is impossible to have a negative power in one of these functions.

With these numbers: image80.png, k>0.

As a>0 and b>0, the product x should always be greater than 0, therefore x>0.

In summary:

a>0, b>0, a≠1, b≠1, x>0

Explain how you arrived at your general statement.

First, the sequences given with observed. After determining that each sequence had a constant exponent, but the bases did not remain constant, I used the change of base formula. This allowed the logarithm to be rewritten in terms of logs written with other bases.

image27.png   then image32.pngimage28.png   then   image28.png


image33.pngimage34.png

After observing the sequences and determining the nth term, the validity of the general formula was tested using other values of a, b, and x. The validity was proved through the formula:


image31.png

After taking the logarithms in base x:

image38.pngimage39.png

Therefore,

image40.png

Then it is necessary to use the change of base formula once again to get the following expression:


image41.png

Since variables were given, it was most likely that substiution would be needed to find the general formula:

image42.png

Consequently, after multiplying both sides by cd, the general formula is:

image43.png

...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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