Logarithms are introduced by John Napier
If N and a (≠1) are any two positive real numbers and for some real number x ,
ax= 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.