- Level: International Baccalaureate
- Subject: Maths
- Word count: 1125
IB Math Portfolio- Topic 1
Extracts from this document...
Introduction
International Baccalaureate
Math Methods SL Portfolio 1:
Logarithms
December 02, 2009
The intent of this portfolio is to explore, investigate, and model the patterns found in logarithmic functions.
In considering the first logarithm in the first sequence:
It is important to know what it represents.
By definition, a logarithm is the inverse of an exponential function.
Therefore, when:
The next expression in the series is:
By definition:
This expression is more difficult to solve as easily.
However, if 8 is seen instead as:
The expression becomes easier to solve
There is a pattern in the bases of each consecutive expression.
In the first set, the base is defined as:
where n is the nth term in the sequence.
In considering the whole sequence:
Sequence 1:
Term 1 | Term 2 | Term 3 | Term 4 | Term 5 |
So, in term 5, the base is
Therefore, in terms 6 and 7 of all sequences, the value n must be equal to 6 and 7 respectively
Term 6 | Term 7 |
The graph below represents the resulting values in sequence 1 as the exponent of the base increases.
Middle
The numerator stays at k, and since the base of the exponent is represented by n, the denominator is also represented by n.
Therefore, the term of the sequence in terms of p/q is
It is important to look at finding the solution to the answer in a different way.
The change of base rule is generally a rule that puts the logarithmic statement into one that is expressed in base 10, so it is easily calculatable on the calculator.
The change of base rule is this:
For
This can easily be proved by considering the solution by definition of a logarithm.
Now, calculate the following, giving your answers in the form
Sequence 2a | |||
Results | 2 | 1.2 |
Sequence 2b | |||
Results |
Sequence 2c | |||
Results |
Sequence 2d | |||
Results |
Figuring out the third expression in the sequence is a matter of adding together the exponents of the base.
In the first example, the bases have a similar root-- namely, 2.
Sequence 2a |
Think of the bases not as separate values, but instead as exponents of 2.
Sequence 2a(Revised) |
From the results above, it is easy to suspect that the third term of the sequence comes from an addition of the exponents of the bases in the two previous terms:
2+3=5
The same can be done with the next three sequences:
Sequence 2b |
Sequence 2b(Revised) |
From the results above, it is seen that the exponent of the third term is the sum of the two base exponents in the terms preceding it.
1+2=3
Sequence 2c |
Sequence 2c(Revised) |
(-1)+(-3)=(-4)
Sequence 2d |
Sequence 2d(Revised |
Conclusion
Then, by taking the inverse once more, the result for the general statement can be reached:
=
When solving for the general statement, and testing for values of a, b, and x, using the change of base formula caused some problems:
By this thought, both x and a cannot be negative, because 10 to the power of any real number equal to a negative number.
The same can be said for b:
x and b cannot be negative.
In graphing the two equations, the asymptotes and limits can be seen as to where the values do not exist.
It is seen in the graph that the x values of the graph are limited to positive values over 0.
in replacing x with 0, it is seen that the value cannot exist, because no integer a or b, (a or b > 0) to the power of another integer (over 0) can equal a value of 0, but can approach it very closely.
The result for the general statement arrives from different methods. In using the change of base formula as well as using a proven inverse rule, adding together two logarithms of different bases is possible.
This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.
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