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# Understanding Antilogarithm: Definition, Properties, and Calculations

Updated: Mar 8

Antilog is an important concept in mathematics that is used to solve exponential equations. The term antilogarithm is derived from the mathematical concept of logarithm, which represents the exponent required to increase the base to obtain a particular value. To understand antilogarithm, we must first understand logarithm.

Let’s explore the concept of antilog in mathematics. We will understand what antilog is and how to calculate it with or without a calculator and will discuss how antilogarithms can be used to solve equations involving logarithmic functions with examples.

## What is Antilogarithm?

Antilog is the inverse operation of taking the log of any number. Such as if you have a log equation of the form y=logb​(x), where x is the actual number and b is the base, then the antilog is an operation that gets you back x from y.

Such as:

Antilog = by

Or

Antilog = antilogb​(y)

To Be Noted:

Here the base b will always be positive but never zero and one (b > 1), because antilog with base 0 is undefined and antilog with base 1 is it doesn’t give meaningful results.

## How to Calculate Antilog without a Calculator?

Before calculating logarithmic notation, you should be well aware of their parts. Logarithmic notation can be calculated by separate expressions in two parts normally called the characteristic and mantissa part.

The characteristic and mantissa parts are two distinct parts that make up a logarithm of a number.

The characteristic part represents the integer part and the mantissa part represents the fractional part of the expression to be calculated. We can calculate antilog without using a calculator by two methods:

### Manual Antilog Calculation Method

Let us suppose a real number 1.4412 to find its antilog. If there is no base to be given then remember that there is always base 10.

Step 1: Write down the logarithmic value and its corresponding base. Here, we have antilog10 1.441.

Step 2: Raise the base (10) to the power of the logarithmic value (1.441). This can be done using a scientific calculator.10^1.441 = 27.618

Step 3: Round the answer to the appropriate significant digits. Hence, antilog (1.4412) = 27.6185

### Using the Antilog Table Method

To find the antilog of 1.441 using the antilogarithmic table, follow the steps below:

Step 1:Extract the characteristic part and mantissa part of the given number.

Characteristic Part = 1

Mantissa Part = .4412

Step 2: Find the row that contains the number(.44) on the left side and step forward to the column of the number(1)at the top of the table.

Step 3: The value located at the intersection of the row and column for .44 and 1 is the antilogarithm value.

In this case, the number is 2761.

Step 4:Find the means difference value from the table. Using the same row (.44) and the fourth number's mean difference column of the mantissa part, which is 2.

The row and mean difference column intersection value is 1.

Step 5: Sum up previously obtained value and this intersects value. 2761 + 1 = 2762

Step 6: Place the decimal point after adding 1 in characteristic part (1+1) digits.

Hence, the antilog 1.4412 = 27.62

## Antilogarithmic Properties

The general properties of antilog are based on the properties of logarithms.These properties are essential for solving complex mathematical problems that involve exponential functions and logarithmic equations. Here are some of the important properties of antilog:

i.  Product of Antilogarithm:

The antilogarithm of the sum of two logarithms is equal to the product of each term's antilogarithm.

antilog (a + b) = antilog(a) × antilog(b).

ii. Quotient of Antilogarithm:

The antilogarithm of the difference of two logarithms is equal to the quotient of the antilogarithm of the first term divided by the antilogarithm of the second term.

antilog(a - b) = antilog(a) ÷ antilog(b).

iii. Power of Antilogarithm:

The antilogarithm of an exponential term is obtained by raising the antilogarithm of the base to the power of the exponent:

antilog(ac) = antilog(c.log(a)).

iv. Inverse of Antilogarithm:

The inverse of the antilogarithm is the logarithm.

y = antilog(x),then x = log(y).

v.  Identity of Antilogarithm:

The antilogarithm of zero is equal to one:

antilog(0) = 1.

vi. Inequality of Antilogarithm:

The antilogarithm of any positive number is always >1.

antilog(x) > 1.

vii. Antilogarithm of a Constant Multiple:

The antilogarithm of a constant times term is the power of the antilog of the number.

antilog (k.a) = (antilog (a))^k.

## Examples of Finding Antilog

Example 1:

If the log of 1000 is 3, then what is the antilog of 3?

Solution:

We know that antilog is the inverse process of log. Let’s elaborate on it.

Log10 1000 = 3

1000 = antilog10 3

Hence, the antilog of 3 is 1000.

Example 2:

What is the Antilog of 5 with base 8?

Solution:

Given Data: antilog (base 8) 5 =?

antilog8 5 = 85

85 = 32,768

The antilog of 5 with base 8 is equal to 32,768.

Example 3:

Evaluate antilog10 0.3102

Solution:

Step 1: From the given number, Characteristic Part = 0 and Mantissa Part = .3102.

Step 2: Locate the intersecting value of the row (.31) and the column (0) in the antilog table which is 2042.

Step 3: Using the mean difference part of the table, figure out the intersecting value of the row (.31) and the characteristic number column (0).

The value of the intersecting is 0 because there is no zero column in the mean difference part of the table.

Step 4: Sum up both values: 2042 + 0 = 2042

Step 5: Place a decimal after 0+1 digits: 2.042

Hence, the antilog of 0.3102 is equivalent to 2.042

## Wrap Up

In this article, we've explored the concept of antilog in mathematics, covering fundamental topics such as its definition, calculation methods, and key properties essential for every math learner. Through solved examples, we show its practical applications.