To determine the logarithm of a number, we need to determine 'characteristic' and 'mantissa'.
Determining the characteristic of a logarithm: Following are the two rules for determining the characteristic of a logarithm.
- Rule 1: The characteristic of logarithm of a number greater than or equal to 1 is one less than the number of digits to the left of the decimal point in the number.
Ex: Consider the logarithm of a number 18.48. The number of digits to the left of decimal point is 2. Thus, the characteristic of log 18.48 is: 2 – 1 = 1.
- Rule 2: The characteristic of logarithm of a number less than 1 is a negative number whose numerical value is one more than the number of zeros immediately following the decimal point.
Ex: Consider the logarithm of a number 0.00245. The number of zeros immediately after the decimal point is 2. So, the characteristic of log 0.00245 is: – (2 + 1) = – 3 = 3.
Determining the mantissa of a logarithm: We determine the mantissa of the logarithm of a number with the help of .
The position of a decimal point in a number is immaterial for finding the mantissa, i.e., log 34, log 3.4, log 0.34, log 0.034, etc. all have the same mantissa. We follow the procedure given below for finding the mantissa.
||Ignore the decimal point in the given number and consider its first four significant figures. In case there are less than four significant figures, add zeros to the extreme right and if more than four significant figures, consider only first four and approximate the fourth with the help of fifth.
||Read log table in a horizontal line lead by first two figures and in the vertical column headed by the third figure. Note down this number.
||Read in the same horizontal line, the number written in the mean difference column headed by the fourth figure and add it to the previous number.
||The sum so obtained gives the required mantissa.
||Using log table, find log 204.3.
||In 204.3, number of digits to the left of decimal point is 3. So, the characteristic of log 204.3 = 3 – 1 = 2. For finding the mantissa of log 204.3 from log table, we proceed in the row headed by 20 and reach the number in the same row under the column headed by 4. The number is 0.3096. To this number, we add the mean difference headed by 3 but in the same row. Thus, mantissa of log 204.3 is 0.3102.
∴ log 204.3 = 2 + 0.3102 = 2.3102.