Definition : We call arithmetic progression, a sequence of numbers such that each of them is equal to the previous one increases or decreases by a constant number called reason.
Example:
Increasing arithmetic progression of reason + 2 :
1 ; 3 ; 5 ; 7 ; 9 ; 11 ; 13 ; 15 ; 17 ; 19 ; ...
Descending arithmetic progression of reason - 3:
19 ; 16 ; 13 ; 9 ; 6 ; 3 ; 0 ; - 3 ; - 6 ; ...
1. 1. 2. - GEOMETRIC PROGRESSION
Definition :We call geometric progression a sequence of numbers such that each of them is equal to the preceding multiplied or divided by a constant number called reason.
Example :
Rising geometric progression of reason :
1 ; 2 ; 4 ; 8 ; 16 ; 32 ; 64 ; 128 ; ...
Decreasing geometric progression of reason 3:
1 ; 1 / 3 ; 1 / 9 ; 1 / 27 ; 1 / 81 ; ...
1. 2. - LOGARITHM OF A NUMBER
1. 2. 1. - LOGARITHM OF A NUMBER GREATER THAN 1
Let's take an increasing geometric progression of reason 10 and the first of which is equal to 1 :
1 - 10 - 100 - 1 000 - 10 000 - ; .... 10n
Now let us take an arithmetic progression of reason + 1 and whose first term is equal to 0.
0 - 1 - 2 -3 -
4 -5 ; .... n
Let's 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 from 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.
We write : log for logarithm :
2 = log 100 = log 102
5 = log 100 000 = log 105
n = log 10n
and especially
0 = log 1
1 = log 10
The logarithm therefore corresponds to the exponent of the base power
;2 is the logarithm of 102 = 100; 3 that of 103
= 1 000 ;5 that of 105 ...
Rule :The logarithm of a power of 10 greater than 1 is positive and equal to the exponent of the power of 10.
Remark : The reason for the geometric progression being 10, logarithms obtained, are called decimal or base 10 or vulgar.
1. 2. 2.- LOGARITHM OF A NUMBER INCLUDED BETWEEN 0 AND 1
The same geometric progression as defined above, gives :
The same arithmetical progression as above to negative numbers, gives :
- n .... ;-5 ; - 4 ; - 3 ; - 2 ; - 1 ; 0 ....
Let's write the two progressions one below the other. As before, the number of the arithmetical progression is the logarithm of the corresponding number of the geometric progression.
Rule:
The logarithm of a power of less than 1 is negative and equal to the exponent of the power of 10.
The same method was used to determine the logarithms of the decimal numbers and logarithm tables were established. We will talk about these tables a little later.
1. 2. 3. - SUMMARY TABLE
Let's summarize what has just been said.
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
We will note that :
Negative numbers have no logarithm ;
The logarithm of 1 is zero ;
The logarithms of numbers greater than 1 are positive ;
The logarithms of numbers greater than 0 and less than 1 are negative; ;
Only the powers of 10 have logarithms of integers.
Examples :log 1 000 = 3 ; log 0,001 = - 3
1. 3.- PROPERTIES OF LOGARITHMS
1. 3. 1.- LOGARITHMS OF A PRODUCT
Let two numbers be A and B, and a and b their logarithms.
a = log A
b = log B
We can write from the very definition of logarithms (paragraph 1. 2.)
A = 10a
B = 10b
Now the product AB:
P =AB =10a . 10b = 10a + b
and taking the logarithm of the two members :
log AB = a + b
= log A + log B
Hence the following basic formula:
log AB = log A + log B
You immediately see the interest of logarithms : we have replaced the calculation of a product by that of a sum.
By generalizing, we will write :
log ABCD = log A
+ log B + log C + log D
Example : We must calculate the logarithm of the number 300.
We will break down 300 into a product of factors less than 100.
300 =3 x 100
log 3 = 0,47712
log 100 =2
Hence, applying the previous rules :
log 300 = log 3 +
log 100
log 300 = 0,47712 +
2
log 300 = 2,47712
We could still have written :
300 =10 x 30
log 10 = 1
log 30 = 1,47712
log 300 = 2,47712
Or :
300 = 3 x 10 x 10
log 3 =0,47712
log 10 = 1
log 10 = 1
log 300 = 0,47712
+ 1 + 1 = 2,47712
1. 3. 2. - LOGARITHM OF A QUOTIENT
Either the quotient A / B = Q
From where we draw :
A = B . Q
Let's apply the previous rule :
log A = log B + log Q
or log Q =log A - log B
Therefore :
A division has been replaced by a subtraction.
Example:
Let the quotient :200 / 10 of which we want to know the logarithm.
We can write :
log 200/ 10 =
log 200 - log 10
Or 200 =100 x 2
from where log 200 = log 100 + log 2 = 2 + 0,30103 =
2,30103
and log 10 = 1
from where log 200 - log 10 = 2,30103 - 1
and finally log 200/ 10 = 1,30103
It is quite obvious that we could have initially calculated the quotient of 200 / 10, or 20, and looked for the logarithm of 20 or computed it according to the known rules and knowing that 20 = 2 x 10.
Another example :
Let the quotient 59/27 of which we want to know the logarithm :
log 59 / 27 = log 59 - log 27
The table or simply a calculator called "scientific" gives us directly the values :
log 59 =
1,77085
log 27 = 1,43136
From where
log 59 / 27 = 0,33949
1. 3. 3. - LOGARITHM FROM A HIGH NUMBER TO A POWER "P"
Let the number A be high at the power p:
Ap =A
x A x A x A x.....x A
Therefore : log Ap = log A + log A +
log A +.....+ log A
log Ap = p log A
Example : What is the logarithm of 1 000 ?
1 000 = 10 x 10 x 10 =103
Let's apply the last relationship :
log 103 = 3 log 10
= 3 x 1
log 103 =3
Another example:What is the logarithm of 102,5 ?
log 102,5 = 2,5 log 10
= 2,5 x 1
log 102,5 = 2,5
Third and final example:
What is the logarithm of 1 600 ?
1 600 = 16 x 100 = 42 x 102
We find ourselves with a product whose every term is 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
is :
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 a root extraction.
Especially if
m / n = 1/ 2, we have a square root extraction :
You feel even more, from now on, the considerable interest which the logarithms can present. Any extraction of root (as complicated as we want) that would be inextricable by the classical method of calculation, resolves very quickly using logarithms.
Let's take an example : Either to extract the square root of 100.
Now, the number R with logarithm 1 is 10, so :
R = 10
We see that logarithms make it easy to perform complex calculations provided, of course, that you know the rules of application.
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Negative numbers have no logarithm ;
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