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Logarithm Bases

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Introduction

Math portfolio

Find the nth term of a logarithmic sequence

In this portfolio I will investigate how we can use changes in the base of a logarithmic sequence to find a general expression. We start with the fallowing logarithmic sequences;

  • image00.pngimage00.png
  • image18.pngimage18.png
  • image30.pngimage30.png

:

:

:

  • image38.pngimage38.png

 We can use the last sequence of the sequences above to show a pattern of the change of base in the sequences. By using this information we can deduce the nth term of any logarithmic sequence. We can also see that the last sequence is written with the base and the number if the logarithm both using m raised to a power, but the number of the logarithm, mk, is the constant in the expression. If we look at the pattern of the other sequences we can see that their number is also constant. The m exponent increases proportionally to the term for each term it is an increase by 1.

Now that we have m both as a base and as the number of the logarithm we can use the change of base rule to simplify the terms:
image48.png

Using the rules for logarithms we get the expression:      image01.pngimage01.png

The use of m both in the base and the number of the logarithm is a smart thing to do. Because the image10.pngimage10.png

...read more.

Middle

2

1

2

25

25

1

2

2

1

25

125

0,666666667

2

3

0,66666667

25

625

0,5

2

4

0,5

25

3125

0,4

2

5

0,4

25

15625

0,333333333

2

6

0,33333333

25

78125

0,285714286

2

7

0,28571429

25

390625

0,25

2

8

0,25

25

1953125

0,222222222

2

9

0,22222222

25

9765625

0,2

2

10

0,2

As you can see the solutions to the term in both forms are exactly the same, in other words, what we have done, writing the terms in the form image19.pngimage19.png, is a correct way of writing the term, because, as said the solutions are identical. As you can see I have continued the sequence to the tenth term, showing that it is correct not only for the calculations shown earlier. The formulas used in these excel sheets can be viewed in attachments 1 and 2 at the back of the portfolio. Let’s move on to the next section.

Applying image32.pngimage32.pngto another pattern and its use

Now let’s try using some of our newly acquired knowledge to solve a problem involving logarithmic sequences similar to those we have just dealt with. Take a look at these four sequences:

  • image33.pngimage33.png
  • image34.pngimage34.png
  • image35.pngimage35.png
  • image36.pngimage36.png

If you look closely at these sequences you will notice that the terms are consecutive, in the three first sequences certain terms have been skipped, and in the last sequence the two first terms have been switched around. Even though these sequences do not seem to have much in common, there is more than meets the eye. It is possible to find the third answer the two first. To get an overview let’s rewrite the terms to the form image19.pngimage19.png.

First sequence:

image37.pngimage37.png

...read more.

Conclusion



Limitations:
The base must be positive and not equal to one, so:
a > 0, b > 0 (because each of them also shows up alone on a logarithm, they can't both be negative).
Also, ab ≠ 1 -> a ≠ 1/b

Finally, the argument (x) must be positive, so:
x > 0


image05.png

image06.png


And we want
image07.png


So I'm going to rewrite (1) and (2) so that I can make appear
image08.png


image09.png

image11.png



Therefore :
image12.png

Now we have found a general statement for the expression logabx in terms of c and d. Where a >0, b>0, ab ≠ 1, c ≠ -d.  We can therefore say that the general statement is image13.pngimage13.png.

A test of the general rule:

A = 1, B = 2, X = 2

log1×2 2 = ((log 2 ÷ log 1) × (log 2÷ log 2)) ÷ ((log 2 ÷ log 1) + (log 2 ÷ log 2))

1 = ((log 2 ÷ 0) × (1)) × ((log 2 ÷ 0) + (1))

1 = (0 × 1) × (0 + 1)

1 = 0 × 1

A can not be 1.

Attachment 1;

number of logarithm

base

log

k

n

k/n

8

=2^1

=LOG(B16;C16)

3

1

=F16/G16

8

=2^2

=LOG(B17;C17)

3

2

=F17/G17

8

=2^3

=LOG(B18;C18)

3

3

=F18/G18

8

=2^4

=LOG(B19;C19)

3

4

=F19/G19

8

=2^5

=LOG(B20;C20)

3

5

=F20/G20

8

=2^6

=LOG(B21;C21)

3

6

=F21/G21

8

=2^7

=LOG(B22;C22)

3

7

=F22/G22

8

=2^8

=LOG(B23;C23)

3

8

=F23/G23

8

=2^9

=LOG(B24;C24)

3

9

=F24/G24

8

=2^10

=LOG(B25;C25)

3

10

=F25/G25

number of logarithm

base

log

k

n

k/n

25

=5^1

=LOG(B3;C3)

2

1

=F3/G3

25

=5^2

=LOG(B4;C4)

2

2

=F4/G4

25

=5^3

=LOG(B5;C5)

2

3

=F5/G5

25

=5^4

=LOG(B6;C6)

2

4

=F6/G6

25

=5^5

=LOG(B7;C7)

2

5

=F7/G7

25

=5^6

=LOG(B8;C8)

2

6

=F8/G8

25

=5^7

=LOG(B9;C9)

2

7

=F9/G9

25

=5^8

=LOG(B10;C10)

2

8

=F10/G10

25

=5^9

=LOG(B11;C11)

2

9

=F11/G11

25

=5^10

=LOG(B12;C12)

2

10

=F12/G12

Attachment 2;

...read more.

This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.

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