- Level: International Baccalaureate
- Subject: Maths
- Word count: 1169
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 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
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
a, b ≠ 1. The best method to check the validity of my general statement is to use different values of a, b and x.
- 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
- 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
- For all the calculations:
CASIO GDC CFX-9850GB PLUS
- 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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