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LOGARITHM math capsule

log ! log ! log!
You know this was the most scary term in mathematics for me when i was in high school!
But when i studied it , it became the most easy concept to learn and i used to do the problems in seconds . And sometimes that too without using pen and paper.

LOGARITHM

If 'a' is a positive real number other than 1 and 'b' is a rational number such that aⁿ= B then we say that logarithm of B to base 'a' is n or 'b' is the logarithm of N to base 'a', written as 

logₐ B = n
So, aⁿ = B <=> log ₐ B = n

e.g. 7º = 1 <=> log ₇ 1 = 0




Understood the definition ?

FUNDAMENTAL LAWS

OF LOGARITHMS -

(1) FIRST LAW :

If m and n are positive rational number, then 
logₐ (mn) = logₐ m + logₐ n

If m1, m2, m3 , m4, ..... mn are positive rational numbers then
logₐ (m1*m2*m3*...*mn) = logₐ(m1) + logₐ(m2) + logₐ (m3) + ..... + logₐ (mn)

(2) SECOND LAW :

If m and n are positive rational numbers then,

logₐ (m / n) =  logₐ m - logₐ n

 (3) THIRD LAW :

If m and n are positive rational numbers, then,
logₐ mⁿ = n logₐ m

(4) FOURTH LAW :

Logarithm of one with any base is always zero i.e. log ₐ 1 = 0 .

(5) FIFTH LAW :

Logarithm of one with any base is always zero i.e.  logₐ a = 1.

(6) SIXTH LAW :

If m is a positive rational number and a, b are positive real numbers such that a ≄ 1 , b ≄ 1 then

logₐ m = logₓ m / logₓ a  

(7) SEVENTH LAW

If 'a' is a positive real number and n is a positive rational number , then 

a logₐ n = n

(8) EIGHTH LAW 

If 'a' is a positive real number and n is a positive rational number , then

logₐ n = logₐ₂ n² = logₐ₃ n₃ = .....


Why i studied this?
Because my exams were up !
Don't tell me , yours too are. 

SOME USEFUL RESULTS-

result 1 

if a > 1, then 
1) logₐ x < 0       if 0< x < 1        2) logₐ x > 0   for x > 1     3) x > y => logₐ x > logₐ y 

result 2

1) logₐ x > 1     if 0< x < a      2) logₐ x = 1    if  x = a     3)  logₐ x < 1    if x = a

4) logₐ x < 0    for all x > 1    5) logₐ x = 0   for x = 1     6) logₐ x > 0   for all x satisfying 0< x < 1

7) x > y => logₐ x <  logₐ y 


result 3

For x > 0, a > 0 ≄ 1 ,            logₐ n(x) = (1 / n) logₐ x

COMMON LOGARITHM AND NATURAL LOGARITHM

Logarithm can have any positive base other than 0 . But there are two most important base which are generally used.

COMMON LOGARITHM
Logarithm to the base '10' is called common logarithm. It is also called brig's logarithm.
e.g. log 100 = log₁₀ 100 = 2

NATURAL LOGARITHM
Logarithm to the base e is called natural logarithm.It is also called Napier logarithm. logₑ x is usually denoted by lnx. e is irrational number between 2 and 3.

ln x = y => eˠ= x

CHARACTERISTICS AND MANTISSA OF A LOGARITHM -

The logarithm of positive real number 'n' consists of two parts 
1. The integral part is known as the characteristics . It is always an integer .
2. The decimal part is called as the mantissa. The mantissa is never negative and is always less than one.

TO FIND THE CHARACTERISTICS 

CASE 1 
The characteristics of the log of a number greater than 1 is positive and numerically one less than the number of digits the decimal part .

CASE 2
The characteristics of log of a number less than 1 is negative and numerically one more than the number of zeroes immediately after the decimal point. It is represented by bar over the digit

LOG COMPLETED HAH! 

ANTILOGRITHM

The positive number 'a' is called the antilogarithm of a number b , is log a = b . If 'a' is antilogarithm of b we write     a = antilog b
So a = antilog b <=> log a = b 

RULES FOR INSERTING DECIMAL POINT

Two rules are used for inserting a decimal point.

Rule 1
When the characteristics of the logarithm is positive , we insert the decimal point after the n+1 digit where n is the characteristics .

Rule 2 
When the characteristics of the logarithm is negative , we insert the decimal point such that the first significant figures is at nth place, where n is the charactertics .

Okay lil' master lets do some exercises !    


logarithm math capsule 1
logarithm 1
logarithm math capsule 2
logarithm 2

logarithm math capsule 3
logarithm 3

logarithm math capsule 4
logarithm 4


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