- Level: International Baccalaureate
- Subject: Maths
- Word count: 777
Type I - Logarithm Bases
Extracts from this document...
Introduction
IB Standard Portfolio Assignment
Type I – Mathematical Investigation
Logarithm Bases
This investigation will determine the relation between different sets of sequences. The sequences include logarithms. This investigation will be tested using technology.
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 exponents are constant. The following two terms of each sequence were determined:
Log28 , Log48 , Log88 , Log168 , Log328 , ,
Log381 , Log981 , Log2781 , Log8181 , ,
Log525 , Log2525 , Log12525 , Log62525 , , ,
, ,,,,
Let’s start with the first sequence (Log28 , Log48 , Log88 , Log168 , Log328 , , ) and determine an expression for the nth term:
1 | 2 | 3 | 4 | 5 | 6 | 7 |
2 | 2 | 2 | 2 | 2 | 2 | 2 |
2 | 4 | 8 | 16 | 32 | 64 | 128 |
The value 2 was used to determine the nth term. is worked out from the table above in the form of
Middle
27
81
243
729
2187
I used the value 3 to determine the nth term since the value of 2 did not work.
is worked out from the table above in the form of by applying the change of base rule therefore:
The numerator and denominator both have the value log3 in them which means they can be canceled out thus leaving us with
Here are two graphs to test its validity and accuracy:
Both graphs are identical towards each other which indicate that both functions of the graphs are the same.
Let’s take the third sequence (Log525 , Log2525 , Log12525 , Log62525 , , ,) into consideration:
1 | 2 | 3 | 4 | 5 | 6 | 7 |
5 | 5 | 5 | 5 | 5 | 5 | 5 |
5 | 25 | 125 | 625 | 3125 | 15625 | 78125 |
I used the value 5 to determine the nth term. is worked out from the table above in the form of by applying the change of base rule therefore:
The numerator and denominator both have the value log5 in them which means they can be canceled out thus leaving us with
Here are two graphs to test its validity and accurateness:
Both graphs are identical towards each other which indicate that both functions of the graphs are the same.
Let’s take the fourth and final sequence ( ,, , , , ) into concern:
1 | 2 | 3 | 4 | 5 | 6 |
m | m | m | m | m | m |
Conclusion
Clarification of How I Arrived at My General Statement:
The sequences given were firstly observed. I determined and have assured that each sequence had a constant exponent but the bases of each one were not. After observing the sequences I then switched the exponents and bases together with the help of the change of base rule. Basically I inversed them:
=
After observing the sequences and determining the nth term we had to test the validity of our general statement using other values of a, b, and x. The validity was proved by finding this formula:
=
This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.
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