# Antilogarithms

Antilogarithm (in short, antilog) is the inverse operation of the logarithm. The logarithm takes a number and returns the exponent to which the base is raised to obtain that number, while the antilogarithm takes an exponent and raises the base to it.

Mathematically, if logbx = y, then by = x and antilogb(y) = x

For example, as the logarithm (base 10) of 100 is 2, the antilogarithm of 2 is antilog(2) = 100.

While computing the natural log’s antilogarithm, we raise ‘e’ to that power. For example, if ln1 = 0, then e0 = 1, and thus, the antilogarithm (base e) of 0 is antiloge(0) = 1.

## Finding

### Using the Antilogarithm Table

We can use the antilog of a number easily using a scientific calculator. However, if we are not provided a calculator, we can find the antilogarithm values of a positive or a negative number using the antilog table given below.

The antilog table is divided into three blocks:

1. The first block is the main column (the first column) with numbers from 0.00 to 0.99
2. The second block (the differences columns) with digits 0 to 9
3. The third block (the mean differences columns) with digits 1 to 9

Now, using the antilog table, let us find the antilogarithm value (base 10) of the number 2.3924.

Step 1: Separating the characteristic part (the integer part) and the mantissa part (the decimal part), we get:

Characteristic part = 2 and mantissa part = 0.3924

Step 2: Now, let us find the corresponding value of the mantissa part 0.3924 from the antilog table.

The corresponding difference value in row number 2, which starts with 0.39, is 2466.

Step 3: The corresponding mean difference value in row number 4, starting with 0.39,  is 2.

Step 4: Adding the obtained values, we get 2466 + 2 = 2468

Step 5: The decimal point is placed in the position obtained by adding 1 to the characteristic part. Here, on adding 1 to the characteristic part 2, we get 1 + 2 = 3

Thus, by adding the decimal point after 3 digits from the left, we get 246.8

Thus, the antilog of 2.3924 is 246.8

### Without Using the Antilogarithm Table

Now, let us find the antilog of 2.3924

Here, the characteristic part = 2, and the mantissa part = 0.3924

Thus, antilog(2.3924) = 102 × 100.3924

Since, 100.3924 ≈ 2.468

⇒ antilog(2.3924) = 100 × 2.468

⇒ antilog(2.3924) = 246.8

## Solved Examples

Find antilog(1.7845) using the antilog table.

Solution:

Here, the characteristic part = 1 and the mantissa part = 0.7845
The corresponding difference value of the mantissa part in row number 0.78 and starting with 4 is 6081
The corresponding mean difference value in row number 5, starting with 0.78, is 7
Now, on adding the obtained values, we get 6081 + 7 = 6088
Since the characteristic part is 1, the decimal point is placed after 2 (= 1 + 1) places from the left.
By  adding the decimal point, we get 60.88
Thus, antilog(1.7845) = 60.88

Find antilog(3.0103)

Solution:

Here, antilog(3.0103) = 103 × 100.0103
Since 100.0103 ≈ 1.024
⇒ antilog(3.0103) = 1000 × 1.024
⇒ antilog(3.0103) = 1024
Thus, antilog(3.0103) = 1024