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- 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
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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 :
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Mantissa usually consist of a four digit number, and it comes after the decimal point.
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Mantissa is a non negative real number , which is less then 1.
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While determining the mantissa, the decimal point of the number has to be ignored.
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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.
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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
= 2.6992.
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
= 1.8632.
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 .9951, although
its value is - 3.0049 )
Example 4 : Find the log of 0.1234.
Characteristic = - 1.
For mantissa, read from the table a number 1234. From the rows, choose 12, and read off from the number under the column 3. The number given in the log tables is 0899. Now read, in the same row, the mean difference under 4. This number is given as 14.
Mantissa = 0899 + 14 = 0913.
Thus log 0.1234 = Characteristic of 0.1234 + Mantissa of 0.1234
= - 1 + 0. 0913
= - 0.9087 or .0913
The log of 1.234 will be 0.0913.
The log of 12.34 will be 1.0913.
The log of 123.4 will be 2.0913.
The log of 1234 will be 3.0913.
To keep consistency, the log of 0.1234 is written as .0913, (although its value is - 0.9087).
The log of 12344 will be as follows : this is a five digit number, so the last that is the fifth digit will have to be rounded off. The fifth digit is 4, which is less than 5. So take the last digit is 0. Thus the mantissa of 12344 will be same as the mantissa for 1234.
log 12344 = 4.0913.
The log of 12346 will be as follows : the last digit 6 is rounded off as 1 and is added to the second last number 4. Thus the last digit becomes 4 + 1 = 5. So we have to find the mantissa for 1235, which is 0899 + 15 = 08914.
log 12346 = 4.08914.
2. Antilogarithm
Antilogarithm is the exact opposite of logarithm of a number.
If x = log b, then antilog (x) = b = .
Remember that antiloga (x) = .
Antilog table for base 10 is readily available. Antilog tables are used for determining the inverse value of the mantissa.
From the characteristic, the position of the decimal point can be determined.
Antilog tables consist of rows that go from .00, . 01, 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 antilog of the numbers after the decimal point, a particular row has to be read off and the mean difference has to be added from the table.
Example 1 : Find the antilog of 2.6992.
The number before the decimal point is 2, so the decimal point will be after the first 3 digits.
From the antilog table, read off the row for .69 and column of 9; the number given in the table is 5000. The mean difference in the same row and under the column 2 is 2. To get the inverse of mantissa add 5000 + 2 = 5002.
Now place a decimal point after the first 3 digits and you get the number 500.2
Thus antilog 2.6992 = 500.2
Example 2 : Find the antilog of – 1.9087
Convert – 1.9087 in bar notation as follows :
characteristic of – 1.9087 = = –2
mantissa of –1.9087 = – 1.9087– (–2) = 0.0913
so –0.9087 = .0913
The number before the decimal point is 0, the number of zeroes after the decimal point is one.
From the antilog table, read off the row for .09 and column of 1; the number given in the table is 1233. The mean difference in the same row and under the column 3 is 1. To get the inverse of mantissa add 1233 + 1 = 1234.
Now place a decimal point before the 1234 followed by a zero and you get the number 0.01234.
Applications
We will now see how logarithms and antilogarithms of numbers are useful for calculations which are complicated or have very large/small numbers.
Example 1 : Find 80.92 * 19.45.
Let x = 80.92 * 19.45
Use the log function on both the sides.
log x = log (80.92 * 19.45)
log (80.92 * 19.45) = log 80.92 + log 19.45 ( from the laws of logarithms)
From the log tables we get log 80.92 = 1.9080, log 19.45 = 1.2889
Thus log (80.92 * 19.45) = 1.9080 + 1.2889 = 3.1969
log x = 3.1969
Now use antilog functions on both the sides.
x = antilog 3.196
From the antilog tables we see that the antilog of 3.1969 is 1573.0.
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