How to use Antilogarithm Table

Antilogarithm is the exact opposite of logarithm of a number. If \(y=\text{log}\ x\), then \(\text{antilog}\ y=x\).

Antilogarithm tables are used to determine the antilogarithm of numbers. Let us now discuss how to find the antilogarithm (with base 10) of numbers.

Let \(x\) be a positive real number. Then, \(\text{antilog}\ x\) can be calculated as follows:

  • Separate the characteristic and mantissa of x.
  • The inverse value of mantissa has to be read from a standard antilogarithm table. Antilogarithm tables consist of rows headed by the values .00, .01, …, .99 and the columns headed by the values 0, 1, …, 9. Beyond these 10 columns, there is another set of 9 columns which give the mean difference. For determining the inverse value of mantissa, the following has to be remembered:
    • While determining the inverse value of mantissa, the decimal point is not ignored.
    • The inverse value of mantissa is calculated for a four-digit number.
    • If the number of digits is more than 4, then it is rounded off to get a four-digit number.
    • Two numbers have to be chosen from a particular row appearing in two particular columns. The first two digits represent the row number, the third digit represents the column number and the fourth digit represents the mean difference column number. If the fourth digit is 0, we take the mean difference as 0.
    • The inverse value of mantissa is the sum of two numbers with a decimal placed after the first digit.
  • \(\text{antilog}\ x=\text{Inverse value of mantissa} \times10^{\text{Characteristic}} \)

 

Let \(x\) be a negative real number. Then, to calculate \(\text{antilog}\ x\), we follow the steps as mentioned above except the first step. Before separating the characteristic and mantissa, we subtract and add a number one greater than the characteristic so that the mantissa becomes positive.

 

Example 1: Find the value of \(\text{antilog}\ 2.6082\).

Solution: Let \(x=2.6082\gt 0\).

To separate characteristic and mantissa:
Characteristic \(=2\) and mantissa \(=.6082\).

To find inverse value of mantissa:
We read from the antilogarithm table a number .60 8 2. In the row headed by .60, choose the numbers in the column headed by 8 and the mean difference column headed by 2. The numbers are 4055 and 2 respectively.

 

  0 1 2 3 4 5 6 7 8 9 Mean Difference
1 2 3 4 5 6 7 8 9
                                       
.59 3890 3899 3908 3917 3926 3936 3945 3954 3963 3972 1 2 3 4 5 5 6 7 8
.60 3981 3990 3999 4009 4018 4027 4036 4046 4055 4064 1 2 3 4 5 6 6 7 8
.61 4074 4083 4093 4102 4111 4121 4130 4140 4150 4159 1 2 3 4 5 6 7 8 9
                                       

 

The sum of these numbers \(=4055+2=4057\).
Therefore, inverse value of mantissa \(=4.057\).
Hence, \(\text{antilog}\ 2.6082=\text{Inverse value of mantissa} \times10^{\text{Characteristic}}=4.057\times10^2=405.7\).

 

Example 2: Find the value of \(\text{antilog}\ (-2.0306)\).

Solution: Let \(x=-2.0306\lt 0\).

Then, \(x=-3+3-2.0306=-3+.9694\).

To separate characteristic and mantissa:
Characteristic \(=-3\) and mantissa \(=.9694\).

To find inverse value of mantissa:
We read from the antilogarithm table a number .96 9 4. In the row headed by .96, choose the numbers in the column headed by 9 and the mean difference column headed by 4. The numbers are 9311 and 8 respectively.

 

  0 1 2 3 4 5 6 7 8 9 Mean Difference
1 2 3 4 5 6 7 8 9
                                       
.95 8913 8933 8954 8974 8995 9016 9036 9057 9078 9099 2 4 6 8 10 12 15 17 19
.96 9120 9141 9162 9183 9204 9226 9247 9268 9290 9311 2 4 6 8 11 13 15 17 19
.97 9333 9354 9376 9397 9419 9441 9462 9484 9506 9528 2 4 7 9 11 13 15 17 20
                                       

 

The sum of these numbers \(=9311+8=9319\).
Therefore, inverse value of mantissa \(=9.319\).
Hence, \(\text{antilog}\ (-2.0306)=\text{Inverse value of mantissa}\times10^{\text{Characteristic}}=9.319\times 10^{-3}=0.009319\).

 

Example 3: Find the value of \(\text{antilog}\ 4.13276\).

Solution: Let \(x=4.13276\gt 0\).

To separate characteristic and mantissa:
Characteristic \(=4\) and mantissa \(=.13276\).

To find inverse value of mantissa:
We read from the antilogarithm table a number .13 2 8 (which is obtained by rounding off the last digit of .13276). In the row headed by .13, choose the numbers in the column headed by 2 and the mean difference column headed by 8. The numbers are 1355 and 3 respectively.

 

  0 1 2 3 4 5 6 7 8 9 Mean Difference
1 2 3 4 5 6 7 8 9
                                       
.12 1318 1321 1324 1327 1330 1334 1337 1340 1343 1346 0 1 1 1 2 2 2 2 3
.13 1349 1352 1355 1358 1361 1365 1368 1371 1374 1377 0 1 1 1 2 2 2 3 3
.14 1380 1384 1387 1390 1393 1396 1400 1403 1406 1409 0 1 1 1 2 2 2 3 3
                                       

 

The sum of these numbers \(=1355+3=1358\).
Therefore, inverse value of mantissa \(=1.358\).
Hence, \(\text{antilog}\ 4.13276=\text{Inverse value of mantissa} \times10^{\text{Characteristic}}=1.358\times 10^4=13580\).