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

Observe the following :

log_{10} 1 = 0.

log_{10} 0.1 = log_{10} 10^{-1} = -1 is written as 1.

log_{10} 0.01 = log_{10} 10^{-2} = -2 is written as 2.

log_{10} 0.001 = log_{10} 10^{-3} = -3 is written as 3.

log_{10} 0.0001 = log_{10} 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: log_{10} 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 )

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