LOGARITHM TABLES - CHARACTERISTIC AND MANTISSA, SOLVED EXAMPLES, EXERCISE

Please study Logarithms before Logarithm Tables,
if you have not already done so.

It is a prerequisite here.

We make use of various Logarithms'
Formulas in solving problems here.

To determine the logarithm of a number, we need
to determine (i) Characteristic and (ii) Mantissa.

Mantissa is found from Tables.

Besides reading tables for the values, we need to
have the knowledge of interpreting the values.

We get that knowledge here.

We study Characteristic and Mantissa here.

We apply the knowledge to solve problems.

Problems are given for practice in exercise with answers.

Practice makes one perfect.

This is especially true for learning of solving
Algebra Problems (Math Problems).
So, take the exercises seriously and
practice solving the problems.

Characteristic and Mantissa of Logarithms : Logarithm Tables

The logarithm to the base 10 is called
common logarithm, as it is commonly used.

Note : In this page we deal with only common logarithms.

Characteristic of Logarithms of numbers > 1 : Logarithm Tables

Observe the following :

• log₁₀ 1 = 0;
  
• log₁₀ 10 = 1;
  
• log₁₀ 100 = log₁₀ 10² = 2;
  
• log₁₀ 1000 = log₁₀ 10³ = 3;
  
• log₁₀ 10000 = log₁₀ 10⁴ = 4;
  

We know that

• single digit number lies between 1 and 10.
  
• two digit number lies between 10 and 100.
  
• three digit number lies between 100 and 1000.
  
• four digit number lies between 1000 and 10000.
  

We can infer the following from the above.

• For single digit number, log(to the base 10) value
  lies between 0 and 1
• For two digit number, log(to the base 10) value
  lies between 1 and 2.
• For three digit number, log(to the base 10) value
  lies between 2 and 3.
• For four digit number, log(to the base 10) value
  lies between 3 and 4.

• log value of a single digit number = 0 + a positive fraction(<1).
• log value of a two digit number = 1 + a positive fraction(<1).
• log value of a three digit number = 2 + a positive fraction(<1).
• log value of a four digit number = 3 + a positive fraction(<1).

The integral part of the log(to the base 10) value of a number is called its Characteristic.

The (positive) fractional part of the log(to the base 10) value of a number is called its Mantissa.

Examples :

• log 3 = 0.4771 Here Characteristic is 0 and Mantissa is .4771
• log 16 = 1.2040 Here Characteristic is 1 and Mantissa is .2040

• Characteristic of a single digit number = 0.
• Characteristic of a two digit number = 1.
• Characteristic of a three digit number = 2.
• Characteristic of a four digit number = 3.
• ...................................so on.
• Characteristic of a n digit number = (n - 1).

Characteristic of a n digit number = (n - 1)

Characteristic of Logarithms of
+ve numbers < 1 : Logarithm Tables

Observe the following :

• log₁₀ 1 = 0.
• log₁₀ 0.1 = log₁₀ 10^-1 = -1 is written as 1.
• log₁₀ 0.01 = log₁₀ 10^-2 = -2 is written as 2.
• log₁₀ 0.001 = log₁₀ 10^-3 = -3 is written as 3.
• log₁₀ 0.0001 = log₁₀ 10^-4 = -4 is written as 4.

We can infer the following from the above.
[Note: when we speak of log value of a number, that number is positive.]

log value of a number(< 1) with

• no zeros after decimal (before a significant digit)
  = 1 + a positive fraction(<1)
• 1 zero after decimal (before a significant digit)
  = 2 + a positive fraction(<1)
• 2 zeros after decimal (before a significant digit)
  = 3 + a positive fraction(<1)
• 3 zeros after decimal (before a significant digit)
  = 4 + a positive fraction(<1)

Characteristic of a number(< 1) with

• no zeros after decimal (before a significant digit) = 1
• 1 zero after decimal (before a significant digit) = 2
• 2 zeros after decimal (before a significant digit) = 3
• 3 zeros after decimal (before a significant digit) = 4
• ...........................................................so on.
• n zeros after decimal (before a significant digit) = ( n + 1 )
  

Characteristic of a number(< 1) with n zeros after decimal (before a significant digit) = ( n + 1 )

Ponts about Characteristic and Mantissa of Logarithms : Logarithm Tables

The following points may be noted.

• The Characteristic is always an integer.
  It may be +ve or -ve or zero.
  
• The Mantissa is never negative. It is always less than one.
  
• To find the Characteristic of a number first see
  whether it is > 1 or < 1.
  If it is > 1, Characteristic = number of digits in the integral part - 1.
  If it is < 1, Characteristic = ( n + 1 ),
  where n is number of zeros after decimal
  (before a significant digit).
• Mantissa is found using Logarithm Tables.
  

While finding Mantissa, the decimal point has no significance.

Example: log₁₀ 4567 = 3.6587

Here, the Mantissa = .6587 is found from Logarithm Tables.

Characteristic = 3 is arrived at, based on the number of digits in 4567.
4567 has 4 digits. ∴ its Characteristic = 4 - 1 = 3.

Mantissa of 4567 = .6587 = Mantissa of 456.7 = Mantissa of 45.67
= Mantissa of 4.567 = Mantissa of .4567 = Mantissa of .04567
= Mantissa of .004567 = Mantissa of .0004567 etc.

Thus while seeing Logarithm Tables for Mantissa,
we have to consider the digits only and not the decimal point.

The location of decimal point plays a significant role,
in arriving at the Characteristic.

Consider the numbers above for which
the Mantissa is the same ( = .6587 )

• Characteristic of 4567 = 3    ∴ log₁₀ 4567 = 3 + .6587 = 3.6587
• Characteristic of 456.7 = 2   ∴ log₁₀ 456.7 = 2 + .6587 = 2.6587
• Characteristic of 45.67 = 1   ∴ log₁₀ 45.67 = 1 + .6587 = 1.6587
• Characteristic of 4.567 = 0   ∴ log₁₀ 4.567 = 0 + .6587 = 0.6587
• Characteristic of .4567 = 1
  ∴log₁₀ 4567 = 1 + .6587 = -1 + .6587 = -0.3413
• Characteristic of .04567 = 2
  ∴ log₁₀ 4567 = 2 + .6587 = -2 + .6587 = -1.3413
• Characteristic of .004567 = 3
  ∴ log₁₀ 4567 = 3 + .6587 = -3 + .6587 = -2.3413
• Characteristic of .0004567 = 4
  ∴ log₁₀ 4567 = 4 + .6587 = -4 + .6587 = -3.3413

Examples in Logarithm Tables

For Solved Examples and Exercise problems
which make use of the knowledge covered here, go to

Problems on Characteristic and Mantissa.