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# Math Portfolio type I Logarithm Bases

Extracts from this document...

Introduction

Lukasz Weclas

Mathematics Standard Level

Portfolio Assignment Type II

Logarithm bases

Lukasz Weclas

December 2009

Introduction                                                                                 03

Write down the next two terms of each sequence                                        03

Find an expression for the nthterm of each sequence and write in form                 04

Calculate the value of given logarithms                                                 06

Describe how to obtain the third answer in each row from the first two answers        08

Create two more examples that fit the pattern                                        08

Find the general statement that expresses logabx                                        09

Test the validity of your general statement using other values of a, b and x        10

Discuss the scope and limitations of a, b and x                                        11

Explain how you arrived at your general statement                                        11

Technology used                                                                        12

Bibliography                                                                                12

Introduction

Logarithm is defined as the exponent that indicates the power to which a base number is raised to produce a given number[1]. In this assignment I shall attempt to investigate the characteristics of sequences of logarithms. As a conclusion, I will try to find the general statement and finally the range and limitations of a, b and x will be considered.

Middle

. The answers will be justified with the use of GDC Casio CFX-9850GB PLUS.

A:

Now I substitute n for any given number, for example 15. Let n=15

To justify my answer, I will use my GDC to check if it is correct

The answer is the same, therefore it is correct.

B:

Now I substitute n for any given number, for example 12. Let n=16

To justify my answer, I will use my GDC to check if it is correct

The answer is the same, therefore it is correct.

C:

Now I substitute n for any given number, for example 8. Let n=8

To justify my answer, I will use my GDC to check if it is correct

The answer is the same, therefore it is correct.

X:

Conducting the same process is not possible in case of sequence X, but the three equations above should prove that calculation would be possible if k, n .

Calculate the value of given logarithms

The next task is to calculate the following sequences given in the assignment, then give my answers in the form , where p, q :

Conclusion

and
a, b ≠ 1. The best method to check the validity of my general statement is to use different values of a, b and x.
1. Let a = 3, b = 9 and x = 729

Next it is necessary to calculate to value of

Later the use of formula

In this example the statement is true, because

1. Let a = 4, b = 6 and x = 1

By the definition it is known that logwq = j, then q = wj, therefore if q=1 then j = 0. In that case log41 = 0, log61 = 0 and log241 = 0. As a conclusion, the argument of a logartihm has to be different from 1.

Discuss the scope and limitations of a, b and x

For statement  to be true two conditions have to be met:

1) a, b, x > 0

2) a, b ≠ 1

Explain how you arrived at your general statement

I arrived at my general statement when I was thinking how to write the expression for nth of each sequence in form . Then I used the two formulas  and . The series of calculations (and final one presented below) lead to the general statement.

Technology used

1. For all the calculations:

CASIO GDC CFX-9850GB PLUS

1. For all the graphic presentation:

Microsoft Office Word 2007

MathType v. 6.5c

Page

[1] http://m-w.com/dictionary/logarithm

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

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