*Logarithms are introduced by John Napier*

*If N and a (≠1) are any two positive real numbers and for some real number x ,*

* **a*^{x}= N , a>0, a≠1, then x is said to be the logarithm of N to the base ‘a’

^{x}= N , a>0, a≠1, then x is said to be the logarithm of N to the base ‘a’

### *It **is written as *

* this is said to be in the logarithmic **form.*

*Logarithms are defined only for positive real numbers.*

*Standard bases of a logarithm**: There are two bases which are used commonly, then any others and deserve special mention. *

*They are base ‘ 10 ‘ and base ‘e’ where ‘e’ is approximated to 2.78*

*Common Logarithms: **Logarithms to the base 10 are called common logarithms and are denoted by * *.*

*Natural logarithms: Nepierian logarithms: **The logarithms of numbers calculated to the base ‘e’ are called natural logarithms or Nepierian logarithms:*

*Constant ‘e’ is an irrational number with an infinite non terminating value of ‘e’ = 2.718*

*John Napier prepared logarithm tables*

*Base ‘e’ used frequently in scientific and mathematical applications. *

*Logarithms to base ‘e’ or * * are often written simply as ‘ln’*

*1. **logarithm of product :-** The logarithm of the product of two numbers is equal to *

*the sum of the logarithms of these two numbers.*

*2. **logarithm of a quotient :-**The logarithm of a quotient of two numbers is **equal to the difference of the logarithms of those two numbers in the order. *

*logarithm of a Power :- **The logarithm of any power of a number is equal ** **to the product of the logarithm of number and the index of the power. *

*The logarithm of any non zero positive number to the same base is unity*

1. *The logarithm of unity to any non zero base is zero.*

*The logarithms of the same number to different bases are different.*

*Characteristic:** the integral part of the logarithm of a number is called characteristic. *

*Mantissa:-** the decimal part of the logarithm of a number is called mantissa. *

*log2=0.3010 , characterstic=0 and mantisa is 0.3010*

*If a number has “ n “ digits the characteristics of its logarithm is “ ( n-1) “.**conversely if the characterstic of the logarthim of a number is n, the number*

*will have ” ” (n+1)digits *

*If in a decimal fraction there are “ n “ zeroes after the decimal point and before the significant number the characteristic of that logarithm is ((**n+1) ̅ )*

*EX: **log**〖**0.005427 has characteristic 3 ̅. **〗*

*The characteristics is always integer. It may be positive or negative or zero**The mantissa is never negative, it is always less than 1.**If a number has both integral and decimal parts to find its characteristic of the logarithm we have to consider the number of digits in its integral part.*