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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

Table of Contents

Table of Contents                                                                        02

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 image00.png                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.

...read more.

Middle

image44.png. The answers will be justified with the use of GDC Casio CFX-9850GB PLUS.

A: image46.png

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

image47.png

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

image48.png

The answer is the same, therefore it is correct.

B: image02.png

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

image03.png

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

image04.png
The answer is the same, therefore it is correct.

C: image05.png

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

image06.png

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

image07.png

The answer is the same, therefore it is correct.

X:image08.png

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 image09.pngimage10.png.

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 image00.png, where p, q image09.pngimage10.png:

image11.png

...read more.

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

image37.png

Next it is necessary to calculate to value of image38.png

Later the use of formula image40.png

In this example the statement is true, because image41.png

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

image42.png

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 image43.png 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 image00.png. Then I used the two formulas image39.png and image44.png. The series of calculations (and final one presented below) lead to the general statement.

image45.png

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

...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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