Antilogarithm, often abbreviated as antilog is the inverse operation of a logarithm. It is used to determine the original number or value that has been raised to a certain power, which is the result of taking a logarithm. The introduction of logarithmic tables in the 17th and 18th centuries facilitated the calculation of antilogarithms. These tables contained precalculated values of logarithms and antilogarithms for various numbers. They were widely used by mathematicians, scientists, and engineers for accurate and efficient calculations before the advent of calculators and computers.
- In 1722, the Swiss mathematician and physicist Gabriel Cramer introduced the term “antilogismus” to describe the inverse of logarithms. Later, in 1748, the Swiss mathematician Leonhard Euler used the term “antilogarithm” in his work “Introduction in Analysin Infinitorum” to refer to the inverse operation of logarithms.
- Today, advanced technology with modern calculators, mathematical software, and programming languages provides built-in functions for calculating antilogarithms and other mathematical operations.
In this article, we will discuss the basic definition, notation, Anti-log table, and how to evaluate Anti-log in detail.
What is antilog?
Table of Contents
Here is the basics of antilogarithm.
| Antilog | Example |
| · In mathematics, anti-logarithm is the exponent to which a base must be raised to obtain a specific number. | · If we have a logarithm value of 2 (log base 10), the anti-logarithm of 2 would be 100, as 10 raised to the power of 2 equals 100.
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| · Anti-logarithm is the process of raising a base to the power of a given logarithmic value, resulting in the original non-logarithmic number.
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· If we have a logarithm value of 3 (log base 2), the anti-logarithm of 3 would be 8, as 2 raised to the power of 3 equals 8.
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| · Anti-log refers to the inverse operation of a logarithm, used to find the original value from its logarithmic result.
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· If we have a logarithm value of 0.5 (log base e), the anti-logarithm of 0.5 would be approximately 1.6487, as e raised to the power of 0.5 is approximately 1.6487.
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Notation of Antilog
The antilog function is commonly denoted by the symbol “Anti-log” or “a*log.” It is used to find the inverse of the logarithm of a given number. Here are four common ways to represent the antilog function:
- Y = Anti-log(x)
- Y = a*log(x)
- Y = 10x
- Y = exp(x)
In these notations, ‘x’ represents the input value for which you want to find the antilogarithm. The antilog function essentially raises the base of the logarithm to the power of ‘x’ to obtain the corresponding antilogarithm.
What is mantissa and the characteristics of antilog?
In the logarithmic representation of numbers, the terms “characteristic” and “mantissa” refer to different parts of the logarithm.
Characteristic:
The characteristic represents an integral part of the logarithm and provides information about the magnitude and sign of the original number. It is the whole number part obtained when expressing a logarithm in standard form.
Example:
In log10(100) = 2
Characteristic = 2
Mantissa:
The mantissa represents the decimal part of the logarithm and contains the fractional component of the logarithmic value. It is the decimal portion obtained when expressing a logarithm in standard form.
Example:
In log10(100) = 2
Mantissa = 0
Together, the characteristic and mantissa form the complete logarithmic representation of a number. In the common logarithmic system (base 10), any number can be expressed as the sum of its characteristic and mantissa: log10(x) = characteristic + mantissa.
How to evaluate the Antilog?
The term antilog could be evaluated with the help of an antilog calculator or following the below steps for the manual calculations.
- Identify the base: Determine the base of the logarithm for which you want to evaluate the antilogarithm. The most common base is 10 (common logarithm), but it can be any positive value.
- Identify the exponent: Determine the exponent or the value within the parentheses of the logarithm. For example, if you have a log (base 10) of 2, the exponent is 2.
- Set up the equation: Write the equation for the antilogarithm. It will be in the form of a base (exponent).
- Substitute the base and exponent: Replace the base and exponent in the equation with the values you have identified. For example, if you have base 10 and exponent 2, the equation becomes 102.
- Perform the calculation: Evaluate the result of the exponentiation. In this case, calculate 102 which equals 100.
- Determine the sign: If the original logarithm was negative, the antilogarithm will be a fraction or a decimal. If it was positive, the antilogarithm will be a whole number or a positive fraction.
- Identify the characteristic: Determine the characteristic of the original logarithm. It is the whole number part that indicates the magnitude and sign of the original number. For example, in a log (base 10) of 100, the characteristic is 2.
- Apply the sign: Apply the sign (positive or negative) determined in Step 6 to the characteristic. If the sign is negative, the characteristic will be negative, and if the sign is positive, the characteristic will be positive.
- Write the antilogarithm: Combine the sign (from Step 8) with the mantissa (the decimal part of the original logarithm, if any) and the characteristic (from Step 7) to form the antilogarithm. For example, if the original logarithm was log (base 10) of 100 with characteristic 2, the antilogarithm is 100.
- Finalize the result: Write down the final value of the antilogarithm, considering any decimal points, fractional parts, or rounding if necessary.
Table of Antilog
Here are tables showing the value of the anti-log function with base 5.
| No | Log – value | Procedure | Anti – Log value |
| 0 | 10 | 1010 | 10000000000 |
| 1 | 9 | 109 | 1000000000 |
| 2 | 8 | 108 | 100000000 |
| 3 | 7 | 107 | 10000000 |
| 4 | 6 | 106 | 1000000 |
| 5 | 5 | 105 | 100000 |
Calculations of Antilog
In this section, we understand the concept of Anti-log with the help of an example in detail.
Example 1:
Determine the anti-log (4.1021).
Solution:
To evaluate the anti-log (4.1021), we need to raise the base of the log (which is usually 10 unless otherwise specified) to the power of (4.1021).
We write mathematically
Anti-log (4.1021) = 10(4.1021)
Step 1:
Distinct the decimal and integer parts of 4.1021.
4.1021= 4 + 0.1021
Step 2:
Raise 10 to the power of 4 using the power operator (^).
10(4) = 10000 → (a)
Step 3:
Evaluate the value of 10 raised to the power of the decimal value 0.1021 using the power rule.
10(0.1021) = 1.2650 → (b)
Step 4:
We multiply that eq (a) and (b). We get
10000 x 1.2650 = 12650.27
Step 5:
We become
Value ≈ 12650
Hence, the anti-log (4.1021) is approximately 12650.27
Example 2:
Find anti-log (1.030).
Solution:
To evaluate the anti-log (1.030), we need to raise the base of the log (which is usually 10 unless otherwise specified) to the power of (1.030).
We write mathematically
⇒Anti-log (1.030) = 10(1.030)
Step 1:
Distinct the decimal and integer parts of 1.030.
⇒1.030= 1 + 0.030
Step 2:
Raise 10 to the power of 1 using the power operator (^).
⇒10(1) = 10 → (1)
Step 3:
Evaluate the value of 10 raised to the power of the decimal value 0.030 using the power rule.
⇒10(0.030) = 1.0715 → (2)
Step 4:
We multiply that eq (1) and (2). We get
⇒10 x 1.0715= 10.715
Step 5:
We become
⇒Value ≈ 11
Hence, the anti-log (1.030) is approximately 10.715
Conclusion
In this article, we have discussed the basic definition of an Anti-log, the notation Anti-log, the definition of mantissa and characteristics, the Anti-log table, and how to evaluate the Anti-log with the help of examples. Furthermore, you can understand Anti-log, anyone can simply evaluate any problem of Anti-log.