Investigation on the Property of Logarithm

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Logarithm base
Introduction
There are two goals in this project. First goal is to generate formula to obtain answer of logarithm, whose anti- logarithm and base is exponent of same number, , by using two or more logarithms which has same number for n and mk. Second goal is to prove how the Fundamental Theorem of Arithmetic, which state that any integer greater than 1 can be written as a unique product of prime numbers, can be applied to the formula which is found in first goal. For the first goal:

Firstly, sequences with logarithms, , will be considered and the formula to obtain answer for this sequence and logarithms will be generated. Secondly, logarithms which are not in sequence, but still have same number for m, will be considered. Then the general statement to obtain an answer will be described, and limitation of this general statement will be discussed by using technology. And then this general statement will be proved. For the second goal:

Firstly, new logarithm will be introduced and then how does value of can be expressed by using c, d and f. Then another logarithm will be also introduced and same step will be followed. After that, general formula will be introduced and then will be proved by using induction. Then how the fundamental theorem of arithmetic can be applied to this general formula will be discussed.

Data and Analysis
These three sequences is going to be considered.
1.) , , , , …
2.) ,,,…
3.) ,,,…
Firstly, the next two terms of each sequence is found.
Base of logarithm is increased exponentially in each sequence. Thus next two terms of each sequence are 1.) ,
2.) ,
3.) ,
Furthermore, anti-logarithm of each sequence can be factorized and shown as exponent. Therefore, general formula of each sequence is 1.)
2.)
3.)
Each sequence has patterns that: base of its logarithm increase exponentially. Anti-logarithm is constant and can be written as exponent. Base of exponent in base of logarithm and anti-logarithm is same. Therefore, these sequences can be generalized like , ,…

And general formula of this sequence is

This can be simplified by using the change of base law, . In this case, base of logarithm is changed to m

For next step, let see how this general formula works for other logarithms which are not in sequence 1.) , ,
2.) , ,
3.) , ,
4.) , ,
Base of logarithm and anti-logarithm have to be written in exponent as first step. For example in 1.), all logarithms is written as , ,

And now, by using general formula, exact value for these logarithms are , ,
Same steps are followed for other questions, and these are the exact value for each logarithm 2.) , , 3.) , , 4.) , ,
By considering these logarithms, it seems like following a pattern which is: If anti-logarithm is same and if base of logarithm is product of two other base of logarithm, its value can be shown as fraction which has product of two logarithms as numerator and sum of two logarithms as denominator. Let show this statement by using variables. Let define, and . So, Logarithm which has base of product of c and d is written as. The pattern which is found has value can be shown as fraction, which numerator has product of c and d, and denominator has sum of c and d.

So far, number of a, b, and x was belongs to natural number ( only 3.) has number belongs to rational). Following table shows that this general statement works when a, b, and x belongs rational and irrational number as well. i.)

ii.)

Table i.) shows actual numbers and answer, and table ii.) shows the equation used. Following bullet points explain each function used in the table * , this is the cell D2 and this means first value, in this case C2, is anti-logarithm and second value, in this case A2, is base of logarithm. * , this is the cell A7, and this means Euler...
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