Understanding of Antilogarithms: Properties, Calculation & Examples

The term “antilogarithm” refers to the inverse operation of a logarithm. In mathematics, exponent-based problems solve with the help of logarithms. The antilogarithm or “antilog” use to reverse this process and find the original number 

Antilogarithms use in many fields, including mathematics, science, and engineering. In mathematics, antilogarithms use to solve equations that involve logarithms. In this article, we will explore antilogarithm with its mathematical expression. We will learn how to calculate antilogarithms with the help of some examples.  

Definition of an Antilogarithms 

In mathematics, an antilogarithm is the inverse operation of the logarithm. If log x = y then, Antilog y = x. The antilogarithm of a number is represented by 10^x. Here x logarithmic value for which you want to find antilog. For example, if the logarithm of a number is 2, then the antilogarithm of that number would be 10², which is equal to 100. 

In other words, the antilogarithm of a number is the exponent that a given base must raise to equal that number. For example, the antilogarithm of 100 in the base 10 is 10, because 10 raised to the power of 2 is 100. 

How to calculate Antilogarithms?

There are different methods to calculate antilog. Here are the most common methods used to find a number’s antilogarithm. 

• By using an antilogarithm table
• By using the antilog calculator 

Antilog by using an antilogarithm table

The systematic method is given below to determine the antilog by using the antilog table:

• Identify the logarithmic value for which you want to find antilog.
• Separate the logarithmic value into the characteristics (Integral part of the logarithm) and mantissa (Decimal part of the logarithm).
• Look up mantissa in the antilog table. 
• Find the first two digits after the decimal point in the first column of the antilog table and the third value of mantissa in the row. Write down the corresponding number of rows and columns from the antilog table. 
• Find the corresponding value of the first two digits after decimal point and the fourth value of the mantissa in the mean difference. 
• Add the obtained value from steps 4 and 5. 
• Decimal move-forward steps (from the left side) are calculated after adding 1 with a given characteristic. 

Antilog by using the antilog calculator

Calculate the antilog value of 10^x. Here x is the given number. The obtained value is the antilogarithm of the given number. Write down the given antilog value & antilog base, click the calculate button in the antilog calculator. The obtained number is the antilog of the given number. 

Properties of Antilogarithm 

The antilog is the inverse process of the logarithm. It allows you to find the original value from its logarithm. Here are some properties of the antilog:

• The Antilog of zero is one. 
• The antilog of a logarithm is the number itself.
• The antilogarithm reverses the effects of the logarithm function. If y = log _a^x, then the antilogarithm of y to the base “a” is given by x = antilog _a^y.
• The antilogarithm can be used to convert a logarithm from one base to another. If y = log _a^x, then the antilogarithm to the base b is given by x = antilog _a^y = Antilog b^y / Antilog b^a).
• The antilogarithm can express in exponential form. If y = log _a^x, then x can be written as x = a^y. 
• The Antilog of the sum of two logarithms is equal to the product of their antilog. If y = log _a^x and z = log _a^w, then x * w = antilog _a^{y + z}.
• The antilog of the difference of two logarithms is equal to the division of their antilog. If y = log _a^x and z = log _a^w, then x / w = antilog _a^{y – z}.

Solved Examples of Antilogarithm 

Here are some examples that may help you to understand Antilog better. 

Example 1:

Calculate the antilog of 0.0224 by using the Antilog table. 

Solution:

Step 1: Split the logarithmic value into the characteristics and mantissa. Here,

Characteristic (integral part) = 0

Mantissa (decimal part) = .0224

Step 2: Look up mantissa in the antilog table. Find the first two digits after the decimal point (.02) in the first column of the antilog table and the third value (2) of mantissa in the row. 

Antilog Table

	0	1	2	3	4	5	6	7	8	9	Mean difference
											1   2   3	4   5   6	7   8   9
.00	1000	1002	1005	1007	1009	1012	1014	1016	1019	1021	0   0   1	1   1   1	2   2   2
.01	1023	1026	1028	1030	1033	1035	1038	1040	1042	1045	0   0   1	1   1   1	2   2   2
.02	1047	1050	1052	1054	1057	1059	1062	1064	1067	1069	0   0   1	1   1   1	2   2   2

Write down the corresponding number of rows and columns from the antilog table. Here the corresponding number is 1052.  

Step 3: Now find the corresponding value of the first 2 digits after decimal point and the fourth value (4) of the mantissa in the mean difference. 

	0	1	2	3	4	5	6	7	8	9	Mean difference
											1   2   3	4   5   6	7   8   9
.00	1000	1002	1005	1007	1009	1012	1014	1016	1019	1021	0   0   1	1   1   1	2   2   2
.01	1023	1026	1028	1030	1033	1035	1038	1040	1042	1045	0   0   1	1   1   1	2   2   2
.02	1047	1050	1052	1054	1057	1059	1062	1064	1067	1069	0   0   1	1   1   1	2   2   2

Step 4: Now add the obtained values from steps 2 and 3. 1052 + 1 = 1053

Step 5: Decimal move forward 1 step from the right side. (By adding 1 in a given characteristic).

Thus, the antilogarithm of 0.0224 is 1.053.  

Example 2:

Calculate the antilog of 0.0224 by using the calculator. 

Solution:

Here, the given value = x = 0.0224

Calculate 10^x

10^0.0224 = 1.0529

Hence, the antilogarithm of 0.0224 is 1.0529.  

Conclusion 

In this article, we have explored antilogarithm with its mathematical representation. We learned different methods to calculate antilogarithms. We covered important properties of antilog in this article.

We did practice some problems of antilog to understand antilog better. Having knowledge of antilog enables you to engage the complex questions of mathematics.