Before understanding anti-logarithms, you should know about logarithms. Many years ago, when the calculator or the computer was not invented, the calculation of larger numbers was too difficult.

In 1614 first time Mathematician John Napier introduced Logarithm and to find a logarithm he published the logarithm table. In 1620, the anti-logarithm table was represented by Jobst Burgi. After these discoveries of the logarithm, the calculation becomes easier.

The concept of anti-logarithm is the same as the concept of the square and square root being inverse of each other. Such that square of 2 is 4 and the square root of 4 is 2. In this article, we are going to discuss the relationship between anti-logarithms and their properties. At last, we will solve some examples.

Table of Contents

**Relation between Logarithms and Antilogarithms**

Today logarithms consider the most useful tool to find an unknown variable that occurs in an exponent (power) such that 10^{p} = 9 we will apply logarithmic laws to this problem. Logarithms are defined as; if b^{x} = z then “x” is called the logarithms of “z” to the base “b” mathematically it is written as log_{b}^{z} = x

Where z > 0 and a > 0 (but not equal to 1) the relation b^{x} = z and log_{b}^{z} = x are equal. For example, 4^{2} = 16 is equal to log_{4}^{16} = 2.

The inverse of a Logarithm is known as **Antilogarithm**. If log y = x then Antilog x = y. Antilogarithm is meaningless without a logarithm. For example logarithms of 21.12 is 1.3247 and the antilogarithms of 1.3247 are 21.12.

**Note:** Logarithms of negative value don’t exist.

**Properties of Logarithms **

We can’t solve complicated logarithms problems without applying logarithms properties so you should remember these Properties of logarithms. Some properties of logarithms are given below;

- log
_{a}(A ×B) = log_{b}A + log_{b}B - log
_{a}(A / B) = log_{b}A – log_{b}B - log
_{b}^{y}× log_{a}^{b}= [log y / log b] × [log b / log a] - log
_{a}^{y}= log_{b}^{y}× log_{a}^{b} - log
_{a}y^{n}= n log_{a}y

**Definition of characteristics and Mantissa **

If you want to find antilogarithms by using an antilog table then you should know about characteristics and Mantissa. Logarithms of any number have two parts;

- Characteristics
- Mantissa

**Characteristics**

In the logarithms of any number, the integral part is called characteristics. If we take the logarithms of any number which is greater than 1 then we get positive characteristics. But if we take logarithms of any number which is less than 1 we get negative characteristics.

**For example**

Log (21.12) = 1.3247

21.12 is greater than 1 so its characteristic is positive. The integral part of Log 21.12 is 1 which is characteristic.

**Mantissa **

In the logarithms of any number, the decimal part is called Mantissa. It contains four digits. It must be positive. In the above example, .3247 is Mantissa.

**Method to find Antilogarithms**

We can find antilogarithms by using the calculator to take (10)^{ x} where x is the given number

**Example 1.**

Calculate the antilogarithm of the number 1.2345

**Solution**

1.2345 is the given number so take 10^{1.2345} = 17.16

The Antilogarithm of the number 1.2345 is 17.16

**Example 2.**

Find the Antilogarithm of the number 1.0457 by using the Anti-logarithm table.

**Solution**

**Step 1:** First identify the characteristic and mantissa. As we know that the integral part is characteristics and the decimal part is the mantissa. Here

Characteristic = 1

Mantissa = .0457

**Step 2:** you can observe that the antilog table is split into three blocks. Take mantissa and see the First 2 digits (.04) in 1^{st} block, 3^{rd} digit (5) in 2^{nd} block, and 4^{th} digit (7) see in 3^{rd} block.

**Step 3: **Now add the value of the 2^{nd }and 3^{rd} block i.e. 1109 + 2 = 1111

**Step 4:** If the characteristic is positive then add 1 to it and if the characteristic is negative then subtract it from 1. Now the characteristic is one so after adding 1 we get 2, put the decimal point after two digits from the right.

Antilog 1.0457 = 11.11

The antilogarithm of different values along with the base can be evaluated with the help of an antilog calculator by Allmath (https://www.allmath.com/antilog-calculator.php) to get the step-by-step solution in no time.

**Example 3.**

Solve 7.4567 × 4.3445 by using a logarithm.

**Solution**

Let

x = 7.4567 × 4.3445

Taking logs on both sides

Log x = log (7.4567 × 4.3445)

Using the logarithm Product rule

Log x = log 7.4567 + log 4.3445

Calculate the log by using the logarithm table. The logarithm table is divided into three blocks. You can look at the decimal point and not need to move so characteristics are zero for both.

So the first 2 digits of decimal places find in the 1^{st} block and 3^{rd} digit of the decimal place find in the 2^{nd} block and 4^{th} digit of the decimal place find in the 3^{rd} block. Then add the value of the 2^{nd }and 3^{rd} blocks. After taking the logarithm we get

Log x = 0.8725 + 0.6379

Log x = 1.5104

Taking Antilog on both sides

Antilog (log x) = Antilog (1.5104)

x = Antilog (1.5104)

By using an antilog table we get

x = 32.39

**Conclusion**

In this article, we have discussed the history of logarithms and Antilogarithms. John Napier and Jobst Burgi were great mathematicians who gave the concept of logarithms and anti-logarithms. The definitions of logarithms and anti-logarithms are defined in this article. All terms related to anti-logarithms are also covered with examples, Such that characteristics and mantissa.

The important laws of logarithms to calculate difficult problems are discussed in this article. We learned in the example section, how to find antilog by using the antilog table with a step-by-step solution. After reading this article you will be able to calculate the Anti-logarithm problems.