FINDING THE LOG VALUE OF A NUMBER

The procedure to find the log value of a number involves three major steps. They are: 1) finding characteristic, 2) finding mantissa, and 3) finding anti-logarithm. The integral part of a common logarithm is called characteristic and the fractional part is called mantissa.  Note that the characteristics can be zero, positive or negative, but the mantissa is always positive. Now let us discuss these three steps in detail.




1. Finding Characteristic: In the first stage, we have to find out the characteristic. As discussed earlier, if the digits in the number are more than one, the will be one less than the number of digits to the left of the decimal place. For example, the characteristic of 415.42 is 2, as the number of digits to the left of the decimal place is 3. Similarly, characteristic of 17.23 is 1 and 7.23 is 0. In the case of the numbers which are less the characteristic is equal to one more than the number of zeros after the decimal point and before any significant digit. Thus, characteristic of 0.98 is −1,10.098 is −2, 0.00908 is −3 so on and so forth.

2. Finding Mantissa: To find out the mantissa of a number, you have to use logarithm table. Logarithm tables are presented at the end of this unit. For example, you want to find mantissa of the number 3451. First you have to look at the log tables at the row corresponding to 34 (the first two digits of the given number) and the column corresponding to 5 (the third digit of the given number). The mantissa is 5378. Now look at the mean difference 1 (the fourth digit in the given number) in the same row. The value is Add this 1 to 5378 to obtain 5379. So, for the number 3451, the mantissa part is 0.5379. You already know that the characteristic is 3 for this number. So the log 3451 is .5379.

Note : that mantissa is always positive. It is not affected by the position of the decimal point. That is to say, the mantissa of   would be the same. Looking at the table, it can be seen that the mantissa value of 245 is 0.3892. The characteristic of a number can be decided upon by looking at the digits in that number itself and the mantissa can be obtained from the table using the first four significant digits. Look at the following table and observe how the characteristic is changing without a change in the mantissa value.

Number         Log Value 

2450.0           2.3892

245.0             3.3892

24.5                1.3892

2.45                0.3892

0.245             1.3892

0.0245           12.3892

0.00245         13.3892

Note:  For some log values, you can find a bar over the characteristic. Putting bar over the characteristic indicates that the part where the bar appeared is negative and mantissa (the decimal part) is positive.

3. Anti Logarithms: As you know the logarithm tables give the value of mantissa in the logarithms of Whereas the antilog tables give the value of the number whose log value is known. Suppose in the above example, log value 3.3892 is known. We are now interested in finding out the corresponding actual number whose log value is 3.3892 the number 2450. Here, we can say that the antilog of 3.3892 is 2450. Now let us learn how this antilog value is found from antilog tables. In order to find the antilog of 3.3892, first consider only the mantissa part, Look at the antilog tables at the row corresponding to .38 and column corresponding to number is 2449. Look at the mean column at 2 in the same row, and the value is 1. By adding 1 to 2449, the digits in the antilog value will be 2450. he next task is to decide the decimal position. In the log value of 3.3892 the characteristic is 3. So according to rules   earlier, there should be four digits in the antilog number. Therefore, place a decimal value after four digits. That means, 2450.0 is the original value. To find the number corresponding to log 2.3892, the digits in antilog value obtained from the table will have to be the same as in the earlier case. Only the position of decimal point will change, which will have to be decided by the characteristic. In this case, characteristic is So according to rules given earlier, the antilog must be less than ‘1’ and there must be one zero after the decimal and before the first significant digit in the result. Thus antilog 2.3892 would be 0.0245.

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