Understanding Antilog: Definition, Properties, Rules, and Calculations

The opposite operation to taking a logarithm is called an antilog. Logarithms play a crucial role in mathematics allowing us to analyze exponential growth or decay, solve exponential equations and handle large numbers. By performing the antilog we can retrieve the original value from its logarithmic form

Antilogarithms find wide-ranging applications across fields such as engineering, economics, statistics, and scientific research benefiting from their ability to reverse the logarithmic process and retrieve original values. 

In this article, we will explain the antilog concept in detail. We will also discuss the steps used to evaluate the antilog of different numbers. We will discuss the properties and rules of antilog along with examples.

Antilog 

The word antilog refers to the process of taking a logarithm in reverse. Furthermore, the antilog (antilogarithm) of a number is the value when raised to a specific base and the exponent that results in the original number.

Antilog is the inverse of the log function. It involves finding the number raised to a given exponent to produce a specified result.

Mathematically if we have a logarithmic expression of the form logₐ(x) = y where a is the base x is the number and y is the exponent then the antilogarithm can be represented as:

antilogₐ(y) = x

In simpler terms if you have the logarithm of a number and you want to find the original number you can use the antilogarithm.

Let’s suppose if log₃ (9) = 2 the antilogarithm of 2 to the base 3 would be 9:

antilog₃ (2) = 9

Steps Used to Determine the Antilog 

The following steps can be used for finding antilog values both using an antilog table and without using a table:

Finding Antilog using an Antilog Table:

• First of all, you’ve to find the log value for which you want to find the antilog.

• Locate the corresponding logarithmic value in the antilog table.

• Look in the same row or column as the logarithmic value to find the antilog value.

• The antilog value in the table represents the original number or argument corresponding to the logarithmic value.

Finding Antilog without using an Antilog Table:

• Identify the logarithmic value for which you want to find the antilog.

• Find the base of the log: For example if it is a common logarithm (base 10) or a natural logarithm (base e).

• Use the formula for the antilog based on the logarithm’s base.

• For common logarithms: antilog(x) = 10x where x is the logarithmic value.

• For natural logarithms: antilog(x) = ex where x is the logarithmic value.

• Calculate the antilog using the formula and the logarithmic value.

• If using a calculator enter the logarithmic value and press the antilog or 10x button for common logarithms. For natural logarithms use the exponential function (ex) button.

• If performing manual calculations use the properties of exponents and the value of the base (10 or e) to calculate the antilog.

Remember to consider any rounding or significant figure rules based on the context or desired accuracy.

Properties of Antilog 

The properties of antilog help govern their behavior and allow for various operations and manipulations.

Here are some important properties of antilog

• Base Matching:

To ensure consistent results the base used in the antilogarithm should match the base used in the logarithm.

Example:

The antilogarithm must likewise be computed using base 10 if the logarithm is taken to that base. Using different bases can lead to incorrect results.

• Inverse Relationship with Logarithms:

The connection between logarithms and antilogarithms is inverse. The original integer is returned when the antilogarithm is applied to the logarithm result. In a similar vein the exponent utilized in the antilogarithm is returned when a logarithm is applied to it.

Example:

logbby=x   

• Exponentiation:

Antilogarithms involve raising the base to a given exponent to obtain the original value. Exponentiation is the key operation in calculating antilogarithms.

Example:

If y = 3 then x = b3 is the antilogarithm of ‘y’ with base ‘b’.

• Addition and Subtraction:

The equivalent original numbers can be multiplied or divided when adding or subtracting in the antilogarithmic form.

Example:

If x = ba and y = bc 

Then for addition 

 x × y = b (a + c)

 and for subtraction 

 x / y = b (a - c).

• Multiplication and Division:

When multiplying or dividing antilogarithms the corresponding original values can be exponentiated.

Example:

If x = ba and y = bc then (x × y) = (ba) × (bc) = b (a + c) and (x / y) = (ba) / (bc) = b (a - c).

• Power Rule:

The power rule allows for raising an antilogarithm to power by exponentiation of the original value to the desired power.

Example:

If x = ba then (xm) = (ba)m = b (a × m).

These properties and rules help manipulate antilog in calculations, conversions and problem-solving. 

Example Section:

Example number 01:

Let’s suppose the value of the log is 4 and base is 10. Evaluate the antilog. 

Solution

We find the antilog of the given function with a step-by-step solution

Step 1:

 Identify the values

Base (b)=8

Log value= 3

Step 2: 

Apply antilog

The following formula can be used to find antilog.

X= by

Now put the value in a given formula

X= 104

X= 10000

Example number 02:

Let’s suppose the 4.3010. Determine the antilog.

Solution:

The base of the logarithm which is typically 10 unless otherwise noted must be raised to the power of 4.3010 to obtain the anti-logarithm of 4.3010.

This has the following mathematical expression:

Anti-log (4.3010) = 104.3010

With the help of the following step we determine our required result. 

Step 1: 

First of all separate the decimal parts and integer parts. 

4.3010 = 4 + 0.3010

Step 2:

 Raise 10 to the power of the integer part (4).

104 = 10000

Step 3: 

Using the exponent rules determine the value of 10 raised to the power of the decimal portion (0.3010):

10(0.3010) = 1.9953

Step 4: 

Multiply the results of steps 2 and 3 together:

10000 x 1.9953 = 19953.28

We get:

Value ≈ 19953.28

Therefore, the anti-logarithm of 4.3010 is approximately 19953.28.

Conclusion 

In this article we have discussed the detailed concept of antilog and also we discussed the steps used to evaluate the antilog of different numbers in detail. Furthermore, we have discussed the properties and rules of antilog.

Moreover, we have discussed the antilog in detail with the help of practical examples. After studying this article anyone can defend this topic easily. 

FAQs of Antilog

Question 1:

What is the application of antilog?

Solution:

Antilog finds applications in scientific calculations particularly when working with logarithmic scales such as pH sound intensity of earthquake magnitudes. They are also used to solve exponential equations and interpret results obtained from logarithmic transformations in data analysis.

Question 2:

Is it possible to calculate antilog with a calculator?

Solution:

Yes, you can calculate antilog using a calculator. Many calculators offer the function for base 10 raised to the power of x, which is used to evaluate the antilog.

Question 3:

Are there different bases for antilog?

Solution:

Yes, there exist various bases for the antilogarithm. In many cases base 10 is used and for exponent the base is used for e also called natural logarithm. Simply this is used to determine the actual value of the given function.