- How to determine logarithm of a given number
If x is a positive real number then log x can be written as
log x = characteristic of x + mantissa of x.
Remember that the base is 10 and we are considering natural logarithms or logs only.
Characteristic of x is an integer that can be either positive or negative depending on whether x > 1 or
0 < x <1.
Mantissa of x has to be read from the log tables.
i) How to determine the characteristic of a logx :
If x > 1, then count the digits on the left of the decimal point; if the number of digits is y, then the characteristic is (y - 1).
If 0 < x < 1, then count the number of zeroes appearing in the right side of the decimal point; if the number of zeros is z, then the characteristic is ñ ( z + 1 ). This is
also written as ( z + 1 ), read as (z + 1 ) bar.
ii) How to determine the mantissa of a logx :
As mentioned earlier, the mantissa has to be read from a standard log table. Log tables consist of rows that go from 10,11, up to 99. The columns have values 0,1, 2, up to 9. Beyond the 10 columns, there is another column which is known as the mean difference. For determining the mantissa, a particular row has to be read off and the mean difference has to be added from the table.
The following has to be remembered :
Mantissa usually consist of a four digit number, and it comes after the decimal point.
Mantissa is a non negative real number , which is less then 1.
While determining the mantissa, the decimal point of the number has to be ignored.
Most of the log tables give values of mantissa up to four digits only. For more than a four digit mantissa, we have to round off the last digit.
Number with same sequence of digits have same mantissa.
Example 1 : Find the log of 500.2.
Characteristic = 2.
For mantissa, read from the table a number 5002. From the rows, choose 50, and read off from the number under the column 0. The number given in the log tables is 6990. Now read, in the same row, the mean difference under 2. This number is given as 2.
Mantissa = 6990 + 2 = 6992.
Thus log 500.2 = Characteristic of 500.2 + Mantissa of 500.2
= 2 + 0.6992
Example 2 : Find the log of 72.98.
Characteristic = 1.
For mantissa, read from the table a number 7298. From the rows, choose 72, and read off from the number under the column 9. The number given in the log tables is 8627. Now read, in the same row, the mean difference under 8. This number is given as 5.
Mantissa = 8627 + 5 = 8632.
Thus log 72.98 = Characteristic of 72.98 + Mantissa of 72.98
= 1 + 0.8632
Example 3: Find the log of 0.0009887.
Characteristic = -4.
For mantissa, read from the table a number 9887. From the rows, choose 98, and read off from the number under the column 8. The number given in the log tables is 9948. Now read, in the same row, the mean difference under 7. This number is given as 3.
Mantissa = 9948 + 3 = 9951.
Thus log 0.0009887 = Characteristic of 0.0009887 + Mantissa of 0.0009887
= - 4 + 0.9951
= - 3.0049 ( The log of 0.009887 is also written as