Created it, 05/10/16
Update it, 05/10/17
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1. 1. - CONCEPT OF PROGRESSION
1. 1. 1. - ARITHMETIC PROGRESSION
Definition: One calls arithmetic progression, a succession of numbers such as each one of them is equal to the precedent increases or decreases by a constant number called reason.
Example:
Increasing arithmetic progression of reason + 2 :
Decreasing arithmetic progression of reason - 3 :
1. 1. 2. - GEOMETRIC PROGRESSION
Definition : One calls geometric progression a succession of numbers such as each one of them is equal to the precedent multiplied or divided by a constant number called reason.
Example :
Increasing geometric progression of reason 2:
Decreasing geometric progression of reason 3:
1. 2. - LOGARITHM OF ONE NUMBER
1. 2. 1. - LOGARITHM OF ONE NUMBER HIGHER A 1
Let us take an increasing geometric progression of reason 10 and of which first is equal to 1 :
Now let us take an arithmetic progression of reason + 1 and whose first term is equal to 0.
Let us write these two progressions one under the other:
|
P.G. : |
1 |
10 |
100 |
1 000 |
104 |
105 |
10n |
|
P.A. : |
0 |
1 |
2 |
3 |
4 |
5 |
n |
If we take two numbers of the same column, we will say by definition that the number of the arithmetic progression is the logarithm of the number of the geometric progression.
Thus, we will say that 2 is the logarithm of 100.
One writes: log for logarithm
The logarithm thus corresponds to the exhibitor of the basic power 10 ; 2 is the logarithm of 102 = 100 ; 3 that of 103 = 1 000 ; 5 that of 105
Regulate: The logarithm of a power of 10 superior with 1 is positive and equal to the exhibitor of the power of 10.
Note: The reason of the geometric progression being 10, the logarithms obtained, are known as decimal or at base 10 or vulgar.
1. 2. 2. - LOGARITHM OF ONE NUMBER RANGING BETWEEN 0 AND 1
Same geometric progression that that higher definite, gives :
Same arithmetic progression that above towards the negative numbers, gives:
Let us write the two progressions one below the other. As previously, the number of the arithmetic progression is the logarithm of the corresponding number of the geometric progression.
|
P.G. : |
1 / 10n |
. |
1 / 105 |
1 / 104 |
1 / 103 |
1 / 102 |
1 / 10 |
1 |
10 |
102 |
|
P.A. : |
- N |
. |
- 5 |
- 4 |
- 3 |
- 2 |
- 1 |
0 |
1 |
2 |
As follows: log 10-5 = log (1 / 105) = - 5 . and log 10-1 = log 1 / 10 = - 1
Regulate: The logarithm of a power of 10 lower than 1 is negative and equal to the exhibitor of the power of 10.
The same process was used to determine the logarithms of the decimal numbers and of the tables of logarithms were established. We will speak about these tables a little later.
1. 2. 3. - SUMMARY TABLE
Let us summarize what has just been known as.
|
Decimal numbers > 0 |
Power of 10 |
Logarithms |
|
0 |
- |
- |
|
0,0001 |
10-4 |
- 4 |
|
0,01 |
10-2 |
- 2 |
|
0,1 |
10-1 |
- 1 |
|
1 |
100 |
0 |
|
10 |
101 |
1 |
|
100 |
102 |
2 |
|
1 000 |
103 |
3 |
|
10 000 |
104 |
4 |
It will be retained that:
The negative numbers do not have a logarithm;
The logarithm of 1 is equal to zero;
The logarithms of the numbers larger than 1 are
positive;
The logarithms of the numbers higher than 0 and lower
than 1 are negative;
Only the powers of 10 have as logarithms of the
integers.
Examples: log 1 000 = 3 ; log 0,001 = - 3
1. 3. - PROPERTIES OF THE LOGARITHMS
1. 3. 1. - LOGARITHMS OF ONE PRODUCT
Are two numbers A and B, and a and b their logarithms.
We can write according to the definition even logarithms (paragraph 1. 2.)
That is to say maintaining product AB:
and by taking the logarithm of the two members:
From where the following fundamental formula :
You see the interest of the logarithms immediately : one replaced the calculation of a product by that of a sum.
While generalizing, we will write :
Example: We must calculate the logarithm of number 300.
We will break up 300 into a product of factors lower than 100.
From where, by observing the preceding rules:
One could still have written:
OR
1. 3. 2. - LOGARITHM Of ONE QUOTIENT
That is to say quotient A / B = Q
From where one draws: A = B . Q
Let us observe the preceding rule :
Thus :
A division was replaced by a subtraction.
Example:
That is to say the quotient: 200 / 10 we want to know the logarithm.
We can write:
It is quite obvious that we could, at the beginning, to calculate the quotient of 200 / 10, that is to say 20 and to seek the logarithm of 20 or to calculate it according to the known rules and knowing that 20 = 2 X 10.
Another example:
That is to say the quotient 59 / 27 of which we want to know the logarithm:
The table or quite simply of a computer known as scientific gives us the values directly:
1. 3. 3. - LOGARITHM OF ONE HIGH NUMBER A ONE POWER P
That is to say number A raised with the power p:
Example: Which is the logarithm of 1 000 ?
Let us apply the last relation:
= 3 X 1
log 103 = 3
Another example: Which is the logarithm of 102,5 ?
log 102,5 = 2,5
Third and last example:
Which is the logarithm of 1 600 ?
We are with a product of which each term is high squared:
From where: log 1 600 = log 42 + log 102
log 42 = 2 log 4
= 2 X 0,60206 = 1,20412
log 102 = 2 log 1
= 2 X 1 = 2
and finally log 1 600 = 1,20412 + 2
that is to say : log 1 600 = 3,20412
1. 3. 4. - GENERALIZATION
The formula above remains valid if (p) is a fractional number: p = m / n
However, a fractional power is an extraction of root.
In particular if m / n = 1 / 2, we have an extraction of square root :
You smell more still, as of now, the considerable interest which the logarithms can present. An unspecified extraction of root (as complicated as one wants) who would be inextricable by the method of calculation traditional, is solved very quickly using the logarithms.
Let us take an example: That is to say to extract the square root of 100.
However, the number R which has as a logarithm 1 is 10, therefore :
We see that the logarithms make it possible to easily carry out complex calculations in condition, of course, to know the implementing rules well.
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