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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:

,     where a>0, a≠1, b>0

The sets of sequences are as follows:

log₂8,  log₄8,  log₈8,  log₁₆8,  log₃₂8

log₃81,  log₉81,  log₂₇81,  log₈₁81

log₅25,  log₂₅25,  log₁₂₅25,  log₆₂₅25

:

:

:

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, ,  log₁₅₆₂₅25

:

:

:

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

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

Middle

25

125

625

3125

15625

78125

3.

Use the change of base formula:

Then apply the rule:

cancels out on both sides

What remains:

Here are two graphs to test its validity and accurateness:

Y=

Y=

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.

Use   to change the base of the logarithm to ten.

cancels out.

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

Justification of this statement using technology:

()

()

()

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

The answer was . This form will be used.

2.

3.

4.

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.

By recognizing that  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.

Conclusion

= error/ not possible

The example applies for a, that b≠1.

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

and

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

With these numbers: , 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.

then    then

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:

After taking the logarithms in base x:

Therefore,

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

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

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

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

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