In Mathematics, the anti-logarithm (also known as the inverse logarithm or exponential function) is the inverse operation of taking a logarithm.
The concept of anti-logarithms is used in a variety of fields, including mathematics, science, and engineering, where it is often used to convert logarithmic data back into its original form. It is also a fundamental concept in the study of exponential functions and their properties.
Moreover, in this article basic definition of anti-logarithm its formula, and methods to calculate it with the help of examples and table will be discussed.
What is anti-logarithm?
The inverse method of determining the logarithm of a given number is to calculate the antilog, commonly referred to as "Anti-Logarithms," of that number. Consider, if y is the logarithm of a number x with base b, then we can say x is the antilog of y to the base b. It is defined by
|
Log bX = Y |
x = antilog y |
For example, if log (base 2) of x = 4, then the anti-log of 4 with a base of 2 is 2^4, which is equal to 16.
In short, the anti-logarithm is the inverse operation of a logarithm, which involves raising the logarithmic base to the power of the given logarithmic value to get the original number.
Formula:
The formula for finding the anti-log of a given logarithmic value is:
anti-log(y) = base^ y
Here, "y" is the given logarithmic value, and "base" is the logarithmic base used in the original logarithmic expression.
How to calculate Anti-log:
Before finding the anti-log first we understand the parts of the Anti-log like characteristics and mantissa part.
Characteristic part:
The characteristics part of an antilog is the integer part of the antilogarithm. It is determined by the number of digits to the left of the decimal point in the original logarithm. For example, the logarithm of 123 is 2.0899, which means that the characteristic part is 2.
The characteristic part of an antilog is important because it helps determine the scale of the final result. For example, if you are using anti-logarithms to solve a problem, you may need to adjust your answer based on the characteristic part.
If the characteristic part is positive, you need to multiply the antilog by the corresponding power of the base. If the characteristic part is negative, you need to divide the antilog by the corresponding power of the base.
Mantissa part:
The mantissa part of an antilog is the fractional part of the antilogarithm. It represents the value that needs to be multiplied by the corresponding power of the base to get the final result. For example, if the antilogarithm is 2.0899, the characteristic part is 2 and the mantissa part is 0.0899.
The mantissa part is important because it determines the precision of the final result. The more digits there are in the mantissa part, the more precise the final result will be. For instance, a mantissa part of 0.0899 on an antilogarithm makes it less accurate than one with a mantissa part of 0.089923.
The way to Find the Antilog of a Number:
Two methods to find the Anti-log of a number
• Using an Antilog Table
• Antilog Calculation
Using an Antilog Table:
An anti-log table is a table that contains the values of anti-logarithms for various logarithmic expressions. Here are the steps to use an anti-log table to solve a problem:
Step 1: Identify the base of the logarithmic expression.
For example, if the logarithmic expression is log (base 10) 2.5, the base is 10.
Step 2: Identify the logarithm of the expression.
In the example above, the logarithm is 2.5.
Step 3: Look up the value of the anti-logarithm in the anti-log table.
Find the row in the table that corresponds to the logarithm and the column that corresponds to the first one or two digits of the logarithm. The value at the intersection of the row and column is the value of the anti-logarithm.
Step 4: Adjust the value if necessary.
If the logarithm is not an exact match for any value in the table, you may need to adjust the value by interpolating between the two closest values in the table.
Antilog Calculation:
Anti-logarithm is the inverse operation of the logarithm. It involves calculating the value that results in a particular logarithmic expression. The process of calculating anti-logarithm involves three parts: the base, the logarithm, and the calculation itself.
• Base:
The base is the number raised to power to produce a logarithmic expression. For example, in the expression log (base 10) 100 = 2, the base is 10.
• Logarithm:
The logarithm is the exponent to which the base is raised to produce a particular number. For example, in the expression log (base 10) 100 = 2, the logarithm is 2.
• Calculation:
To calculate antilog, you need to raise the base to the power of the logarithm. For example, to find the anti-logarithm of log (base 10) 2.5, you would raise 10 to the power of 2.5, which is approximately equal to 316.228.
Table:
Here's a table showing some common values of the anti-logarithm function (with base 10):
|
Logarithmic Value |
Method |
Anti-Logarithmic Value |
|
0 |
100 |
1 |
|
1 |
101 |
10 |
|
2 |
102 |
100 |
|
3 |
103 |
1000 |
|
4 |
104 |
10000 |
|
5 |
105 |
100000 |
|
6 |
106 |
1000000 |
|
7 |
107 |
10000000 |
|
8 |
108 |
100000000 |
|
9 |
109 |
1000000000 |
To use this table, simply find the row corresponding to the given logarithmic value and read off the corresponding anti-logarithmic value from the second column.
For example, if you have a logarithmic value of 3.5 in base 10, you can find the corresponding anti-logarithmic value by finding the row with a logarithmic value of 3, and then interpolating between the anti-logarithmic values for 3 and 4. In this case, the anti-logarithmic value would be approximately 3162.278.
Sample Example:
In this section, with the help of an example, the topic is explained.
Example:
Find the antilog of 4.3010
Solution:
To find the anti-logarithm of 4.3010, we need to raise the base of the logarithm (which is usually 10 unless otherwise specified) to the power of 4.3010.
This can be expressed mathematically as:
Anti-log (4.3010) = 10^4.3010
To calculate this value, we can use a calculator or we can use the following steps:
Step 1: Separate the integer and decimal parts of 4.3010:
4.3010 = 4 + 0.3010
Step 2: Raise 10 to the power of the integer part (4) using the exponent operator (^):
10^4 = 10,000
Step 3: Calculate the value of 10 raised to the power of the decimal part (0.3010) using the exponent laws:
10^0.3010 = 1.9953
Step 4: Multiply the results of steps 2 and 3 together:
10,000 x 1.9953 = 19,953.28
We get:
Value ≈ 19,953.28
Therefore, the anti-logarithm of 4.3010 is approximately 19,953.28.
Summary:
In this article, the basic definition of Anti-log its formulas, and basic rules are discussed. Moreover, with the help of a table and example topic is explained. After a complete understanding of this article, anyone can defend this topic.
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