Technology used 12
Bibliography 12
Introduction
Logarithm is defined as the exponent that indicates the power to which a base number is raised to produce a given number^{}. In this assignment I shall attempt to investigate the characteristics of sequences of logarithms. As a conclusion, I will try to find the general statement and finally the range and limitations of a, b and x will be considered.
Write down the next two terms of each sequence
The given examples of logarithms are as follows:
It can be easily noticed that in each sequence every term has the same number of logarithms, respectively:
A: 8
B: 81
C: 25
X: mk
What is changing in the sequences of logarithms are the bases of logarithms. However the similarity presents, as they can be all written down as an=logxnY. In the given data the value of x is respectively:
A: 2
B: 3
C: 5
X: m
Knowing this two thing, I can now calculate and write down the next two terms
of each sequence:
A: a6 = log26 8 = log648, a7 = log27 8 = log1288
B: a5 = log35 81 = log24381, a6 = log36 81 = log72981
C: a5 = log55 25 = log312525, a6 = log56 25 = log1562525
X: a5 = logm5 mk, a6 = logm6 mk
Find an expression for the nth term of each sequence and write in the form
Using the knowledge gained in previous task an expression for the nth term of each sequence can be calculated:
A: an = log2n 8
B: an = log3n 81
C: an = log5n 25
X: an = logmn mk
One of the possible ways to present these expressions in the form ,where p, q Z, is to use formulas (for base substitution) and . The answers will be justified with the use of GDC Casio CFX9850GB PLUS.
A:
Now I substitute n for any given number, for example 15. Let n=15
To justify my answer, I will use my GDC to check if it is correct
The answer is the same, therefore it is correct.
B:
Now I substitute n for any given number, for example 12. Let n=16
To justify my answer, I will use my GDC to check if it is correct
The answer is the same, therefore it is correct.
C:
Now I substitute n for any given number, for example 8. Let n=8
To justify my answer, I will use my GDC to check if it is correct
The answer is the same, therefore it is correct.
X:
Conducting the same process is not possible in case of sequence X, but the three equations above should prove that calculation would be possible if k, n .
Calculate the value of given logarithms
The next task is to calculate the following sequences given in the assignment, then give my answers in the form , where p, q :
The given sequences are similar to the ones given in previous task. I will use the first example whether the formula (conclusion from previous task) can be used here.
D:
The formula can be used in this calculations, therefore I can proceed.
E:
F:
G:
Describe how to obtain the third answer in each row from the first two answers
From the calculations above two conclusions can be drawn necessary to obtain the third answer in each row from the first two answers. Firstly, the argument of logarithm is the same in each sequence (64, 49, 125 and 512 for D, E, F for G respectively). Secondly, it can be noticed that third base of logarithm is a product of the first two (example E: 7 × 49 = 343). Therefore a formula can be created for a1 being the base of first logarithm, a2 being the base of second logarithm and Q being argument of logarithm, such as . As a confirmation, I shall use example E:
But in the given task, there is one more possibility of obtaining the third answer in each row from the first two answers. It can be noticed that numerator of the third answer is a product of first and second answers, when its denominator is a sum of first and second answer. I shall use the same example to present it:
Create two more examples that fit the pattern
To study the pattern deeply it is vital to create two more examples that will show whether or not the formula deduced by me is true.
H:
∴
I:
Find the general statement that expresses logabx
To find the general statement I will let logax = c and logbx = d. The conclusion withdrawn from this mathematical investigation indicate that numerator of the third answer is a product of first and second answers, when its denominator is a sum of first and second answer. As a logical continuation the general statement presents itself:
Test the validity of your general statement using other values of a, b and x
The definition of logarithms provides us with the assumptions that a, b, x > 0 and
a, b ≠ 1. The best method to check the validity of my general statement is to use different values of a, b and x.

Let a = 3, b = 9 and x = 729
Next it is necessary to calculate to value of
Later the use of formula
In this example the statement is true, because

Let a = 4, b = 6 and x = 1
By the definition it is known that logwq = j, then q = wj, therefore if q=1 then j = 0. In that case log41 = 0, log61 = 0 and log241 = 0. As a conclusion, the argument of a logartihm has to be different from 1.
Discuss the scope and limitations of a, b and x
For statement to be true two conditions have to be met:
1) a, b, x > 0
2) a, b ≠ 1
Explain how you arrived at your general statement
I arrived at my general statement when I was thinking how to write the expression for nth of each sequence in form . Then I used the two formulas and . The series of calculations (and final one presented below) lead to the general statement.
Technology used
 For all the calculations:
CASIO GDC CFX9850GB PLUS
 For all the graphic presentation:
Microsoft Office Word 2007
MathType v. 6.5c
http://mw.com/dictionary/logarithm