Logarithm Bases - 3 sequences and their expression in the mth term has been given. All of these equations will be evaluated on a step-by-step basis in order to find an expression for the nth term.

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Mathematics Internal Assessment

LOGARITHM BASES

Abhishek Puri

IB Mathematics-SL

29/05/2009

Mr. Adam Scharfenberger

Part- 1

3 sequences and their expression in the mth term has been given. All of these equations will be evaluated on a step-by-step basis in order to find an expression for the nth term. The technology which will be used to execute this investigation will be a TI-83 graphing calculator and Logger Pro graphing  Program. All mathematical calculations have been made by this graphing calculator and is mentioned where some calculations have not been shown.

(A)

Let us observe the first row of the sequence.

The following sequence is given as:

 

Each of the terms in this sequence are of the form . If analyzed, it can be easily observed that the log base (b) of each term in this sequence is a power of 2 while A remains constant at 8.

Therefore, we can state that each term for the 1st row has the general expression log2n 8, where n is the nth term of the sequence. Now, we must apply this expression in the form  .

log2n 8

=            [Applying change of base formula]

=           [Substituting 8 as a power of 2]

=  

Now, the expression  , should provide us with the values of each term in the sequence.  Thus, to prove  as the general expression for the first row of sequences:

un =

First term of the sequence =

         = log2n 8

(Taking the first term as u1 and so on,)  

L.H.S                        R.H.S

u1 =                            log21 8

u1 = 3                         log2 8 = 3

L.H.S = R.H.S. Hence, verified that  is the general expression for the 1st row of the given sequence in the form   .

Similarly, other terms in the sequence were also verified by using the TI-83 where:

U2:   log48 (1.5)

U3:   = log88 (1)

U4: :   = log168 (0.75)

U5: :   = log328 (0.6)

U6: :   = log648 (0.5)

U7 : :   = log1288 (0.42)

 

Below is the graph for the reciprocal function f(x)=

Now, below is the Graph emphasizing values of given terms of the sequence.

The graph above is a result of plotting f(x)=  and plotting the values of each term in the sequence [x| x≥0] and [y|y≥0]. For our purposes, this graph is confined to the positives as the points in red( which represent the values of the respective terms in the sequence) are positive values.  From the graph, we can observe that the y-value of the co-ordinates plotted. These are in fact, the values of logarithmic terms while the x value is the term of the sequence. For example,

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To find that value of logarithmic term u4= log16 8,

Through applying the change of base formula in the Graphing Calculator we get,

 = 0.75  

(4,0.75) is plotted on the plane and it is also point on the graph, f(x) =  . Thus, in order for a logarithm to be a part of the sequence of the first row, its value along with the term number(n) (where the value is a y co-ordinate and (n) is the x-coordinate) must be a point on the graph f(x) =  

Thus, the next two terms of the sequence have been verified ...

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