Logarithm and antilogarithm are related terms. The two terms are inverses of each other. Logarithms are used in many fields of science, engineering, mathematics, statistics, engineering, astronomy, etc. to manipulate large numbers to avoid complex and complicated calculations.

The antilogarithm, essentially, is the reverse process of calculating the logarithm of a given number. It is also termed as Anti-Log. In instances where both the logarithm and antilogarithm share a base of 10, the transition to a natural logarithm and antilogarithm is achieved by multiplying the values by the conversion factor 2.303. This ensures a seamless adjustment to the base 2.7183, facilitating compatibility between logarithmic operations with different bases.

The concept of antilogs plays a crucial role in visualizing how changes in exponential quantities will affect subsequent quantities. It assists in making useful informed decisions regarding the project’s trajectory.

In this article, we will discuss the concept of the antilogarithm. We will elaborate on its definition, the important steps to find the antilogarithm, and solve some examples.

What is Antilogarithm?

An antilog is a mathematical function that takes an input in the form of a real number and returns a negative logarithm of it. Antilog is the inverse of the logarithmic function i.e.

If log x = y

Then,

x = antilog (y) …………(i)

So, by using the log formula, a logarithmic equation can be converted into an exponential equation.

log x = y => x = 10^y ………..(ii)

Parts of Antilog

If we take the log of a number. It has two parts. Before going to compute the antilog of a number, we will introduce the important terms.

Characteristic:

The characteristic of a logarithm of a number is defined as the whole part of a number that is positive for a number > than 1, while it is negative for a number < 1.

Consider,

b = a x 10ⁿ where 1 ≤ a ≥ 10.

The exponent (index) of 10 is the characteristic of log b. It is important to note that

The characteristic (whole part) of the logarithm of a number > 1, The characteristic of a logarithm is positive and is precisely one less than the number of digits in the integral part of the original number.

The characteristic of the logarithm of a number < 1, is negative and it is one more than the number of zeros instantly after the decimal point of a number.  

Mantissa: 

The mantissa of the logarithm of a number defined as the decimal part of a number that is always positive, is called the mantissa of the logarithm of the original number.

How to Find the Antilog:

Now we will discuss the most important methods of computing the antilog of a number.

·       Using antilog table

·       Using Calculator

Using Antilog Table:

It can be observed that the antilogarithmic table comprises three parts.

·       Observe only the mantissa (decimal part) and ignore the whole part of the logarithm of the number.

·       Consider 1^st two digits of the mantissa and look them in the first block that is pointed with red color. Observe the 3^rd digit of the mantissa in the second block which is pointed with yellow color and the 4^th digit of the mantissa will be looked in the 3^rd  mean difference block which is pointed out with green color.

·       Observe the row corresponding to 1^st two digits of the decimal part (mantissa)

·       Locate the column corresponding to the 3^rd digit of the mantissa until it intersects the corresponding located row of 1^st digits. Note this value.

·       Now this value will be added with the number at the intersection of this value’s row and the mean difference column corresponding to the 4^th digits of the column. Now only the decimal point is to be inserted.

1.      If the characteristic is positive i.e. the whole part of the logarithm of the number > 1, then its numerical value increases by 1 and therefore gives the number of figures to the left of the decimal point in the required number.

2.      If the characteristic is negative i.e. the whole part of the logarithm of that number < 1, then its numerical value decreases by 1 and gives the number of zeros to the right of the decimal point in the required number.

Using calculator:

It is very convenient to find out antilog with the help of using an antilog calculator. Simply you have to write the value of the number as an exponent of 10^. Executing this you will get the desired result. e.g. Antilog(2.1645)=10^2.1645=146.05.

Examples:

Example 1.

Using the antilog table, find the number whose logarithm is 2.0354.

Solution:

Step 1. Separate characteristic and mantissa part.

characteristic=2, mantissa=.0354

Step 2. To find the corresponding value of 1084, locate the row that begins with .03 and move to column number 5 in the given data set. The intersection of this row and column will reveal the desired value of 1084.

Step 3. The number at the intersection of this row and the mean difference column corresponding to 4 is 1.

Step 4. Sum up these values obtained in step 1 and step 2, we obtain the value 1084 + 1 = 1085

Step 5. Since the characteristic is 2. Its numerical value increased by 2 (as there should be three digits in an integral part) and therefore inserting a decimal point to the designated place which is fixed after three digits from left in 1085.

Hence,

Antilog (2.0354) = 108.5 Ans.

Example 2.

Compute an antilog of 2.0354 using a calculator.

Solution:

Step 1. Writing this value as an exponent of 10^.

Antilog (2.0354) = 10^^2.0354 = 108.5 Ans.

Wrap Up:

In this article, we have addressed the concept of the antilogarithm. We elaborated on its definition, and significant methods to find the antilog. In the last section, we have solved some examples in order to apprehend the methods to compute the antilog of a number. Hopefully, apprehending this article you will be able to tackle the problems about antilogarithms.