Understanding Antilog: Definition, Methods, and Examples

Understanding Antilog: Definition, Methods, and Examples:- Logarithms play an important role in solving complex problems but understanding their inverse operation “antilog” is also important. Antilog is short for “Antilogarithm”; and is the inverse function of logarithms While logarithms help find the power to which a given base must be raised to produce a specific number; antilog reverses this process; providing the actual number given the base and its logarithm.

In this article, we’ll delve into the concept of antilog, explore the methods to calculate, and shed light on the examples of these methods.

What is antilog?

It represents the process of finding the original number from its logarithm. In other words, if a logarithm is signified as log_b (x) = y, the antilog of y with base b, denoted as antilog (y) produces the original value x. It is the undoing of the logarithm.

Antilogarithm Notation and Representation:

Antilogarithms are represented in many ways depending on the context and mathematical notation used. The standard representation is antilog_b (y);

Where:

• b = Represent the base.
• y = The logarithm value.

Additionally, antilog can be represented as 10^y when the base is 10; as e^y when using the natural logarithm (base “e”), or as b^y when expressing it w.r.t a specific base “b”.

Before going on with the table-based calculation method; it is important to have a complete understanding of the antilog table.

What is Antilog Table?

To find the antilog of a given number without the need for a calculator, an antilog table is a reference table used. It provides a quick and suitable way to find the original value of a number that has undergone a logarithmic transformation.

Blocks of antilog table:

1. The first block (main column) contains numbers from .00 to .99; representing the decimal part of the logarithm.
2. The second block (differences columns) displays the digits from 0 to 9; which will be added to the decimal part of the logarithm.
3. The third block (mean differences columns) shows the digits from 1 to 9; which are used to adjust the values in the difference columns.

Methods to calculate the Antilog:

Calculation of antilog manually is a valuable skill; especially in those situations when we have no access to calculators or antilog devices is limited. Two main methods to calculate antilog manually are the following:

1. Method 1: Using the Antilogarithm Table.
2. Method 2: Using the Base and Logarithm Formula.

Method 1: Using the Antilogarithm Table.

The characteristic and the mantissa. The logarithm of any number can be represented as the sum of its characteristic and mantissa. The characteristic can be either positive or negative; while the mantissa is always positive.

When dealing with negative numbers; we must take extra care to ensure that the mantissa remains positive. To do this; we add and subtract 1 from the characteristic; which effectively changes the sign of the logarithm.

Steps to calculate the Antilogarithm by table:

• Step 1: Determine the whole number part (characteristic) and the decimal part (mantissa) from the given logarithm.
• Step 2: Focus uniquely on the mantissa. Use the first two digits after the decimal point as the row number and the 3rd digit as the column number in the antilog table. Find the corresponding number in the table.
• Step 3: In the same row, locate the mean difference corresponding to the 4^th digit of the mantissa. Add this mean difference to the number obtained in Step 2.
• Step 4: Position a decimal point immediately after the first digit of the result obtained in Step 3.
• Step 5: Multiply the number obtained in Step 4 by 10 raised to the power of the characteristic, and this final result represents the antilog of the given number.

Method 2: Using the Base and Logarithm Formula.

The formula to find the antilog is:

Antilog(y) = b^y

Here, ‘y’ is the logarithm of the number, and ‘b’ is the base of the logarithm.

Let’s go through the steps to find the antilog using the base and logarithm formula:

Step 1: Identify the logarithm and its base.

Consider you have a logarithm “y” and its base ‘b’, such as log_b y.

Step 2: Take the base ‘b’ to the power of the logarithm “y”.

Calculate b^y using a calculator or by hand.

Step 3: The result of b^y is the antilog of “y”.

The obtained value is the antilog of the given logarithm ‘y’ with the base ‘b’.

Examples of the Antilog:

To gain more understanding of the methods to calculate antilog, let’s solve some examples.

Example 1:

Find the antilog of log₂ 8. (By using Base)

Solution:

Step 1:

Identify the logarithm and its base.

In this case,

y = 8.

b = 2

Step 2:

Take the base ‘b’ to the power of the logarithm ‘y’.

2⁸ = 256

Step 3: The result of 2⁸ is the antilog of 8.

The antilog of log₂8 is 256.

You can also try an antilogarithm calculator to find antilog of given values with steps in a fraction of a second.

Note:

Remember, that when working with antilogs, the base ‘b’ should be the same as the base used in the original logarithm to obtain the correct result.

Example 2: (with table)

Calculate the antilog of 1.3247

Solution:

Using an antilog table.

Step 1:

Determine the whole number part (characteristic) and the decimal part (mantissa) from the given logarithm.

Characteristic: 1

Mantissa: 0.3247

Step 2:

Focus uniquely on the mantissa. Use the first two digits after the decimal point as row number 32 and the 3rd digit as column number 4 in the antilog table. Find the corresponding number in the table.

Row number = 32

Column number = 4

Corresponding value = 2109 + 3

Step 3:

In the same row, locate the mean difference corresponding to the 4^th digit of the mantissa. Add this mean difference to the number obtained in Step 2.

= 2109 + 3= 2112.

Since the characteristics are 1 it is increased by 1 (because there should be two digits in an integral part) and therefore the decimal point is fixed after 2 digits.

Step 4:

Position a decimal point immediately after the two digits of the result obtained in Step 3.

= 21.21

Hence, the antilog of 1.3247 is 21.12.

Wrap up:

In this article, we discussed the concept of antilog, which is the inverse function of logarithms. It helps find the original number from its logarithm. We explored the use of an antilog table, which is a reference table to find antilogs without a calculator.

Understanding Antilog: Definition, Methods, and Examples