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# 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

5(35)

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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