- Level: International Baccalaureate
- Subject: Maths
- Word count: 1960
Logarithms. In this investigation, the use of the properties of logarithms will be used to identify patterns, relationships, and limits of logarithms and sequences.
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Introduction
Introduction
In this investigation, the use of the properties of logarithms will be used to identify patterns, relationships, and limits of logarithms and sequences. A logarithm is an exponent in which the base must have in order to receive the number. All values found will be in the form p/q to further show a relationship with the terms.
Part 1
In the first sequence given, when examined closely, you will see that the log number remains the same throughout the part of the sequence given. You will also notice that it is the base of the logarithms that is altered.
log28, log48, log88, log168, log328, …
To begin, it would prove to be useful to figure out what sort of sequence it really is. By looking that the logarithms, it is safe to conclude that the sequence given cannot be an arithmetic sequence; the reason being that the bases of each logarithm are different, therefore, there is no common difference that describes the sequence as arithmetic.
Since it has been proven that the sequence cannot be arithmetic, the only type of sequence left, is a geometric sequence. A geometric sequence is a sequence whose consecutive terms are multiplied by a fixed, non-zero real number called a common ratio. Therefore, to prove that the sequence is indeed, geometric, the common ratio must be found.
Middle
log34 / log35
4log3 / 5log3
4/5
6
log72981
log36(36)
log34 / log36
4log3 / 6log3
4/6 = 2/3
As the sequence shows, the general expression used to obtain the answer is 4/n.
Value of 5th termValue of 6th term
4/5 4/6
4log3 / 5log3 4log3 / 6log3 multiply log3 to both sides
log34 / log35 log34 / log36 change multiplier into exponent
log81 / log243 log81 / log729 expand exponential
log24381 log72981 change the base to rewrite logarithm
Third Sequence
nth term | Corresponding logarithm | Rewrite into exponential form | Change the base | Change the exponent into a multiplier of logarithm | Simplify ratio |
1 | log525 | log51(52) | log52 / log51 | 2log5 / 1log5 | 2/1 = 2 |
2 | log2525 | log52(52) | log52 / log52 | 2log5 / 2log5 | 2/2 = 1 |
3 | log12525 | log53(52) | log52 / log53 | 2log5 / 3log5 | 2/3 |
4 | log62525 | log54(52) | log52 / log54 | 2log5 / 4log5 | 2/4 = 1/2 |
5 | log312525 | log55(52) | log52 / log55 | 2log5 / 5log5 | 2/5 |
6 | log1562525 | log56(52) | log52 / log56 | 2log5 / 6log5 | 2/6 = 1/3 |
As the sequence shows, the general expression used to obtain the answer is 2/n.
Value of 5th termValue of 6th term
2/5 2/6
2log5 / 5log5 2log5 / 6log5 multiply log3 to both sides
log52 / log55 log52 / log56 change multiplier into exponent
log25 / log3125 log25 / log15625 expand exponential
log312525 log1562525 change the base to rewrite logarithm
Fourth Sequence
nth term | Corresponding logarithm | Rewrite into exponential form | Change the base |
Conclusion
a = 1 b = 1 x = 125
logabx
= log1125 substitute in the values for a, b, and x
= log125 / log1 change-of-base to be able to see the work of the equation easily
=log125/ 0 solve log1, which equals to 0.
Yet another limitation of cd/c+d = logabx is that ab = 0. This cannot be because log0 is not possible. If…
a = 0 b = 5 c = 125
logabx
= log(0)(5)125 substitute in the values for a, b, and x
= log0125 combine
= log125 / log0 change-of-base to be able to see the work of the equation easily.
Not possible since log0 does not exist.
The next limitation of cd/c+d = logabx is that x>0. It is a limitation for the same reasons as the first limitation: a logarithm cannot be negative.
The last limitation on cd/c+d = logabx is that x = 0. This limitation is also a limitation for the same reasons as the second limitation: a logarithm cannot equal zero.
Therefore, from the information provided, you can conclude that the general statement works only with positive numbers.
Conclusion
In conclusion, from the information in this investigation, we can say that the value of the logarithmic sequences in the form of p/q can be reached through the equation k/n. Regularly, the value of logarithmic sequences can be reached through the equation logabx. We also have determined what the general statement was to the expression logabx. The general statement is cd/c+d which has a few limitations that are ab>0, ab = 1, ab = 0, x>0, and x = 0.
This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.
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