Logarithms
If a is a positive real number other than 1 and a x = m , then x is called
the logarithm of m to the base a, written as loga m. In loga m , m
should be always positive.
(i) If m < 0, then loga m will be imaginary and if m = 0, then
loga m will be meaningless.
(ii) loga m exists only, if m , a > 0 and a ≠ 1.
Types of Logarithms
1. Natural (or Napier) Logarithms The logarithm with base ‘ e ’
( e = 2 .718) is called natural logarithms.
e.g. loge x , loge 25 etc.
Note The another way of loge x can be represented as ln.
2. Common (or Brigg’s) Logarithms The logarithm with base 10 is
called common logarithm.
e.g. log10 x , log10 75 etc.
Note In a logarithmic expression when the base is not mentioned,
it is taken as 10.
Characteristic and Mantissa of a Logarithm
The logarithm of positive real number n consists of two parts.
(i) The integral part is known as the characteristic. It is always
an integer positive, negative or zero.
(ii) The decimal part is called as the mantissa. The mantissa is
never negative and is always less than one.
Method to Find the Characteristic
Case I When number is greater than 1.
The characteristic is one less than the number of digits in
the left of decimal point in the given number.
If a is a positive real number other than 1 and a x = m , then x is called
the logarithm of m to the base a, written as loga m. In loga m , m
should be always positive.
(i) If m < 0, then loga m will be imaginary and if m = 0, then
loga m will be meaningless.
(ii) loga m exists only, if m , a > 0 and a ≠ 1.
Types of Logarithms
1. Natural (or Napier) Logarithms The logarithm with base ‘ e ’
( e = 2 .718) is called natural logarithms.
e.g. loge x , loge 25 etc.
Note The another way of loge x can be represented as ln.
2. Common (or Brigg’s) Logarithms The logarithm with base 10 is
called common logarithm.
e.g. log10 x , log10 75 etc.
Note In a logarithmic expression when the base is not mentioned,
it is taken as 10.
Characteristic and Mantissa of a Logarithm
The logarithm of positive real number n consists of two parts.
(i) The integral part is known as the characteristic. It is always
an integer positive, negative or zero.
(ii) The decimal part is called as the mantissa. The mantissa is
never negative and is always less than one.
Method to Find the Characteristic
Case I When number is greater than 1.
The characteristic is one less than the number of digits in
the left of decimal point in the given number.
