Antilogarithm 

Antilogarithm is the inverse operation of a logarithm, used to find the original number from its logarithmic value. If ${\log }_{b}x=y$, then $x=b^y$ is the antilogarithm of y. It plays a key role in scientific calculations, exponential growth models, and competitive exams. Understanding logarithm and antilogarithm concepts simplifies complex multiplications and helps in accurate data interpretation across physics, chemistry, engineering, and finance.

1.0Antilogarithm Meaning

The antilogarithm (often written as antilog) of a number is the inverse operation of a logarithm.

If:

${\log }_{b}x=y$

Then:

$x=b^y$

Here, x is the antilogarithm of y to the base b.

So, in simpler terms:

Antilogarithm tells us what number was originally taken to get a given logarithm.

2.0Antilogarithm Formula

To find the antilogarithm, use this formula:

${\mathrm{Antilog}}_b(y)=b^y$

Where:

• b is the base of the logarithm (usually 10 or e)
• y is the logarithmic value
• $b^y$ gives the original number (antilog)

3.0How to Calculate Antilogarithm

There are three main ways to calculate an antilogarithm:

1. Using the Formula

${\mathrm{Antilog}}_{10}(2.3010)=10^{2.3010}\approx 200$

2. Using a Scientific Calculator

Enter the log value, press the "SHIFT" or "2nd" key, then press the "log" key to get the antilog.

3. Using an Antilogarithm Table

• Split the number into its characteristic (integer part) and mantissa (decimal part).
• Use the antilogarithm table PDF to find the value corresponding to the mantissa.
• Multiply by $10^{\text{characteristic }}$ to get the result.

4.0Rules of Antilogarithm

Here are some important rules and properties:

1. Inverse of Logarithm:

${\mathrm{Antilog}}_b\left({\log }_b(x)\right)=x$

2. Antilog of Sum:

$\mathrm{Antilog}(a+b)=\mathrm{Antilog}(a)\times \mathrm{Antilog}(b)$

3. Antilog of Difference:

$\mathrm{Antilog}(a-b)=\frac{\mathrm{Antilog}(a)}{\mathrm{Antilog}(b)}$

4. Base 10 Antilog:

Commonly used base in logarithmic tables and competitive exams is 10.

5.0Antilogarithm Examples

Example 1: Find the antilog of 2.3010

Solution: 

$\mathrm{Antilog}(2.3010)=10^{2.3010}\approx 200$

Example 2:

If ${\log }_{10}x=-1.523$, find x

Solution: 

$x=10^{-1.523}\approx 0.030$

Example 3:

Solution: 

Use antilog table to find antilog of 0.4771

• Mantissa = 0.4771
• From table: antilog(0.4771) ≈ 3
• Hence, $\text{ Antilog }(0.4771)\approx 3$

6.0Applications of Antilogarithms in Science

• Physics: To compute values in decibel scales, exponential growth/decay, and wave intensity.
• Chemistry: In pH calculations and reaction rates.
• Biology: Population modeling and genetics.
• Engineering: In calculations involving logarithmic amplifiers and signal processing.
• Astronomy & Earth Sciences: Earthquake magnitude (Richter scale), star luminosity.