In this article, we will discuss what is antilogarithm, how to calculate it and how to find the value of the unknown using the logarithm. So, let us get started.

What is Anti logarithm?

 

An antilogarithm is also called antilog and as the name implies it is the inverse of the logarithm of a number.

 

Consider a number "m" having a base "b" and logarithm "n". The mathematical notation of the log and antilog of a number is given below:

If log_b {m} = n, then m = antilog n

How to Find the antilogarithm of a number?

You should be aware of the characteristic and mantissa part of a number before proceeding to find the antilog. These two parts are explained below:

  • Characteristic part: The whole number part of any number is known as the characteristic part. Remember that the characteristic of the logarithm of a number that is greater than one is positive and is one unit lesser than the total number of digits present on the left side of the decimal point.
  • Mantissa part: It refers to the decimal part of the logarithm of a number. It should be greater than 0, i.e. it must be a positive number. When a mantissa part is a negative number, we convert it into a positive one.

Methods for Finding Antilog of a Number

There are two methods for calculating the antilog of a number. Both methods will yield the same result. In this section, we will discuss how to calculate the antilog of the numbers using the two methods.

1. Using the Antilog Table

Follow these steps to calculate the antilog using the antilog table:

Step 1 - Segregate the characteristic and mantissa part of a number. For instance, consider the number 2.1894. The characteristic part is 2 and the mantissa part is 0.1894.

Step 2 - Using the mantissa part, locate the corresponding value in the antilog table. For antilog of the number 3.1894, locate the row that starts with .18, then a column for 9. The corresponding value we got is 1545.

Step 3 - Now, we will look at the mean difference column. Use the same row number, i.e. .18 and find the corresponding value in column 4. In this example, the corresponding value is 1.

Step 4 - Add the corresponding values obtained in steps 2 and 3. In this example, after adding we will get the number 1546.

Step 5 - The last step of this method is to add a decimal point. For this, you need to add 2 to the characteristic part of the number. After adding 1, we get the number 3. Add the decimal point after 3 digits in the number 1546. Hence, the value of the antilog is 154.6.

2. Using Calculator

Another way to calculate the antilog of a number is by using a calculator.  Follow these steps to use this procedure:

Step 1 - Determine the base. When you are calculating the antilog of a number, you always assume that it has the base ten.

Step 2 - Calculate 10^{3.1894} using the calculator. The answer is 154.6.

No matter which method you use for computation of the antilog of the number, the result will be the same. However, using the calculators, you can quickly compute the value and save time.

 

Cologarithm

The logarithm of the inverse of a number is known as the cologarithm. In other words, we can say that the cologarithm of a number is the opposite of its logarithm. Mathematically, it is denoted as:

co log N = log \frac{1}{N} = -log N

 

In the next section, we will solve some examples in which we will compute the antilogarithms of the numbers.

Example 1

Find antilog if log_3 {27} = 3.

Solution

We know that if  log_b m = n, then m = antilog n.

Anti log_3 {3} = 3^3 = 27

 

Example 2

Using the logarithm, find the value of x for the following equation:

x = \sqrt[5] {493}

Solution

Take log on both sides of the equation:

log {x} = log \sqrt[5] {493}

log {x} = \frac{1}{5} log 493

To calculate log 493, you can simply use the log button in the calculator:

log {x} = \frac{1}{5} 2.6928

log {x} = 0.5386

We will isolate x on left hand side of the equation and take log to the right hand side. The log function will turn into an antilog function after changing the side:

x = anti log 0.5386

Use the calculator or the antilog table to calculate the antilog of the number 0.5386. The calculator is the fastest way to do so. Simply take 10^{0.5386} to calculate the antilog.

x = 3.456

 

Example 3

Using the logarithm, find the value of x for the following equation:

x = \sqrt[3] {271}

Solution

Take log on both sides of the equation:

log {x} = log \sqrt[3] {271}

log {x} = \frac{1}{3} log 271

To calculate log 271, you can simply use the log button in the calculator:

log {x} = \frac{1}{3} 2.4329

log {x} = 0.8109

We will isolate x on left hand side of the equation and take log to the right hand side. The log function will turn into an antilog function after changing the side:

x = anti log 0.8109

Use the calculator or the antilog table to calculate the antilog of the number 0.8109. The calculator is the fastest way to do so. Simply take 10^{0.8109} to calculate the antilog.

x = 6.469

Example 4

Using the logarithm, find the value of x for the following equation:

x =\frac { \sqrt[4] {0.3688}} {11.356^2}

Solution

Take log on both sides of the equation:

log {x} = log\frac { \sqrt[4] {0.3688}} {11.356^2}

We will employ logarithmic quotient rule here:

log x = log \sqrt[4] {0.3688} - log {11.356^2}

log x = \frac{1}{4} log 0.3688 -2log 11.356

log x = \frac{1}{4} (-0.4332) - 2 (1.0552)

log x=-0.1083 - 2.1104

log x =  -2.2187

Move the logarithm function to the right to convert it into an antilog function:

x = antilog -2.2187

Insert 10^{-2.2187} in the calculator to compute the antilog:

x = 0.00604

 

Example 5

Using the logarithm, find the value of x for the following equation:

x =\frac { 425 \cdot \sqrt{2.73}} {\sqrt[3] {48.4}}

Solution

Take log on both sides of the equation:

log {x} =log \frac { 425 \cdot \sqrt{2.73}} {\sqrt[3] {48.4}}

Apply the quotient rule of logarithmic functions here. The division sign between the expressions will turn into a subtraction sign using this rule:

log {x} =log (425 \cdot \sqrt{2.73} - log {\sqrt[3] {48.4}}

Using the calculator, find the logarithms of the numbers.

log {x} = 2.6284 + \frac{1}{2} \cdot 0.4362 - \frac{1}{3} \cdot 1.6848 = 2.2849

Converting the log function into an antilog function by isolating x on the left side of the equation will give us the following value:

x = anti log 2.2849 = 192.71

 

Example 6

Using the logarithm, find the value of x for the following equation:

x = \sqrt[3] {105}

Solution

Take log on both sides of the equation:

log {x} = log \sqrt[3] {105}

log {x} = \frac{1}{3} log 105

To calculate log 105, you can simply use the log button in the calculator:

log {x} = \frac{1}{3} 2.0211

log {x} = 0.6737

We will isolate x on left hand side of the equation and take log to the right hand side. The log function will turn into an antilog function after changing the side:

x = anti log 0.6737

Use the calculator or the antilog table to calculate the antilog of the number 0.6737. The calculator is the fastest way to do so. Simply take 10^{0.6737} to calculate the antilog.

x = 4.7173

 

 

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Emma

I am passionate about travelling and currently live and work in Paris. I like to spend my time reading, gardening, running, learning languages and exploring new places.