Antilogarithm and logarithm are related concepts. Logarithms are employed to tackle a lot of numbers to avoid difficult and complex problems. A famous Scottish mathematician whose name was John Napier first used the term logarithm in the 16^th century. 

The Greek word logos has several meanings, one of which is ratio. For the purpose of creating a logarithm table, he introduced e as the base. Henry Briggs, a professor, then created a base 10 logarithm table.  In the year 1620 AD, a Swiss mathematician named Jost Burgi created the table for the antilogarithm.

In this article, we will address the concept of the antilogarithm. We will explore its definition and important methods that will assist in determining the antilogarithm. We will also give some examples to understand the calculations.

Defining Antilogarithm (Antilog):

If log_a x = y, then x = antilog y is the antilogarithm (Antilog) of a number. Another approach to express it is as x = log_a^-1 y. The antilogarithm of the same number is the reverse working process that is used to determine a number’s logarithm.

To compute the antilogarithm, it is now important to comprehend both the terms characteristic and mantissa. The characteristic and mantissa make up the two parts of a number’s logarithm.

Characteristic:

The integral component of a number that is positive for numbers more than 1 and negative for numbers less than 1 is described as the characteristic of the logarithm of a number.

Consider the equation b = a * 10ⁿ, where 1 ≤ a < 10 and the property of log b is its index(power) of 10. A number greater than one has a logarithm whose characteristic is +ve and whose number of digits in the integral component of the original integer is one less.

Mantissa:

The decimal part of a number that is always + ve is referred to as the mantissa of the logarithm and this component of the number is called the mantissa of the number.

Logarithm of a number	Characteristic	Mantissa
log 37.43 = 1.3568	1	.5732
log 63928 = 3.8954	4	.8057
log 0.00574 = 2.5527	– 3	.7589

Techniques to Determine Antilog:

Now we will discuss two important techniques for determining a number’s antilog in this section that are given below:

• By using antilog table
• By using online calculators

Using Antilog Table to Determine Antilog:

An antilog table is divided into three important sections, as shown in Fig.

• Ignore the characteristic and simply take into consideration the decimal component (mantissa).
• In the first block, which is colored in turquoise, we will focus on the first two digits of the mantissa. The 3^rd digit of the mantissa will be observed in the 2^nd block which is highlighted by the green (teal) color. The 4^th digit will be seen in the 3^rd mean difference block, which is indicated by the yellow color.
• Focus on the row that represents the first two digits of the mantissa.
• Find the column that corresponds to the third digit of the mantissa and follow it until it crosses the row of first digits that corresponds to that column. Recall this number.
• This value will now be added to the number that lies at the point where its row and the mean difference column cross, which corresponds to the column’s fourth digits.

Now, all that is to do is to decide the place of the decimal point.

1.  The number of digits to the left of the decimal point in the required number is determined by the numerical value of the characteristic which rises by 1 if it is + ve.
2. If a characteristic is – ve its numerical value is reduced by 1 and the required number is then given with the appropriate number of zeros to the right of the decimal point.

Using Calculators to Determine Antilog:

It is relatively simple to determine antilog using online calculators like an antilog calculator offered by Allmath. The logarithmic value of a number needs to be expressed simply as a 10 exponent. By doing this, you will get your desired answer. E.g. Antilog (1.8026) = 10^1.8026 = 63.479

Examples:

Example 1:

What will be the number whose logarithm is 2.0348 using the antilog table?

Solution:

Step 1: Given data:

Logarithm of the number = 2.0348

Here,

Characteristic = 2, mantissa = .0348

Step 2: Pinpoint the 4^th column and the 1^st row that both begin with “.03”. You’ll receive the number 1081 at the right place as can be seen in the following fig.

Step 3: The number that lies at the position where this row and the mean difference column of 8 intersect is 2.

Step 4: By adding the values from steps 2 and 3 together, we get the number 1081+ 2 = 1083.

Step 5: As the characteristic is 2, its numerical value becomes 2 (as an integral part should have three digits), thus a decimal point was added in the prescribed position, which is fixed after three digits from the left in the number 1083.

Hence antilog (2.0348) = 108.3 Ans.

Example 2: 

What will be the antilog of 1.0348 using the calculator?

Solution: 

Step 1: Write the given value in the exponent of 10^.

Antilog (1.0348) = 10^{^1.0348} = 108.3 Ans.

Wrap Up:

In this article, we have addressed the concept of the antilogarithm which is also termed as antilog. We have elaborated on the fundamental terms logarithm, antilogarithm, characteristics, mantissa, etc. We have explored the important techniques that are useful to determine the antilog or to find a number whose logarithm is given. 

In the last section, we solved some examples that will assist in apprehending basic computations. Hopefully, reading this article you will be able to tackle the problems of logarithm, antilogarithm, etc.