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The reasoning behind conducting this investigation is to identify patterns in logarithmic sequences. Furthermore, after identifying the patterns, one must produce a general statement expressing the general trend within the sequence.

Extracts from this document...

Introduction

International baccalaureate

Mathematics SL

Portfolio Type one

Logarithm bases

CANDIDATE NAME                : Shilpi Singhvi

CANDIDATE NUMBER        :

SCHOOL NAME                        : GANDHI MEMORIAL INT’L SCHOOL

SCHOOL CODE                        : 000902

SESSION                                : MAY 2010

Logarithm Bases

INTRODUCTION

The reasoning behind conducting this investigation is to identify patterns in logarithmic sequences. Furthermore, after identifying the patterns, one must produce a general statement expressing the general trend within the sequence. Once forming a general statement, one must be able to rewrite logarithms in the form of a fraction. This investigation tests the acquired knowledge of logarithms by testing the general statement using various values, forming limitations, and proving scopes. In addition, further investigation may be conducted through the use of graphs, tables of values, and charts.

This assignment is looking for a formula that will give the nthterm for the sequence.

This table gives the next two terms of each sequence.

nth

sequence(1)        

sequence(2)         

sequence(3)

sequence(4)

1

logimage08.png8

logimage09.png81

logimage80.png25

logimage142.pngmimage36.png

2

logimage147.png8

logimage163.png81

logimage171.png25

logimage175.pngmimage36.png

3

logimage10.png8

logimage19.png81

logimage27.png25

logmimage36.png

4

logimage41.png8

logimage44.png81

logimage51.png25

logimage61.pngmimage36.png

5

logimage81.png8

logimage88.png81

logimage95.png25

logimage106.pngmimage36.png

6

logimage126.png8

logimage136.png81

logimage140.png25

logimage141.png mimage36.png

7

logimage143.png8

...read more.

Middle

 mimage36.png= k/6

Thus the nth term is image176.png where p is k and q is n.

Describing how we obtain logimage21.png(64) from logimage26.png(64) and logimage12.png(64).

In order to obtain the answer of the third logarithm, the product of the first and second logarithms must be divided by the sum of the answers of the first and second logarithms.

logimage177.png(x), logimage12.png(x)        logimage81.png(x)image02.png

We use the formula:

logimage144.pngb = image178.png

logimage26.png(x) =          logimage14.png4 = image11.png[1]image03.png

logimage12.png(x) = image13.png         logimage14.png8 = image15.png[2]image03.png

Add [1] and [2]:

logimage14.png4 + logimage14.png8 = image16.png + image17.png

And we have:

logimage18.png32 = image20.png

Therefore the formula we obtain:

logimage21.png(x) =image22.pngorlogimage23.png(x) =image24.png

By replacing the value of x = 64 we get,

logimage21.png(64) = image25.png

logimage26.png(64) = 3 & logimage12.png(64) = 2        (using GDC)

Thus, logimage21.png(64) = image28.png

Describing how we obtain logimage29.png(49) from logimage30.png(49) and logimage31.png(49).

Thus by using the above formula above:

logimage23.png(x) =image24.png

logimage29.png(49) =logimage31.png(49) logimage30.png(49)

                   logimage31.png(49) + logimage30.png(49)image04.png

Solving the equations,

logimage30.png(49) = 1

logimage31.png(49) = 2

Therefore,logimage29.png(49) =image32.png

Describing how we obtain log image33.png(125) from logimage34.png (125) and logimage35.png (125).

Thus, by using the formula

logimage23.png(x) =image24.png,

logimage35.png (125)   = image37.pngimage38.png

image37.png + image38.pngimage05.png

Solving the equations

log image33.png(125) & logimage34.png (125) by change of base log

image39.png= -3image40.png= -1

Therefore,

...read more.

Conclusion

image73.png(x) and used log10ab = log10 a + log 10 b in the denominator.

Now by simplifying loga  x= c and logb x = d  by changing the base logs by the formula above and making loga  x and logb x as the subject of the formula and substituting in the first part we get the general statement as image138.pngimage138.png.

2nd way:

We simplified the statement logab x by making the bases similar.

Now in the equations loga  x= c and logb x = d  we used the formula to make the bases similar i.e. image139.pngimage139.png and making loga x and logb x as the subject of the formula.

Then we substituted it in the simplified version of logab x . By substituting the values we get the general statement.

Through this general statement we can find out the value of the third term in a sequence when the first two terms are given.

Conclusion

In conclusion, after applying the given knowledge of logarithms, a general statement was formed to prove the validity of various sequences. Also through the use of technology, general statements were proven and theories were confirmed.

...read more.

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