Antilog Table

The antilog table is the mathematical tool used to find the antilogarithm of a logarithmic value. The inverse of the logarithmic function is the antilog, that is, if log a = b, then a = antilog b. If log₁₀(a) = b, then a = antilog(b) = 10^b.

For example, antilog(5) = 10⁵ = 100000
antilog(2.354) = 10^2.354 = 226

Antilog tables can be categorized into 3 main blocks. The main column is the first block that shows values from .00 to .99, representing the decimal parts.

The second block is the difference column, which has the digits from 0 to 9. The third block is the mean difference column, which has the digits from 1 to 9. 
 

The antilogarithm is used to reverse the effect of a logarithm and retrieve the original number. The logarithmic value can be divided into two parts: characteristic and mantissa. The integer part is the characteristic, and the decimal part is the mantissa. Thus, the log can be written as: Log (number) = characteristic + mantissa. The characteristic can be positive or negative, but the mantissa should always be positive.

Let’s learn how to use an antilog table to find the antilog of a number, using the example 2.4681. 

Step 1: Identify the characteristic and mantissa
In 2.4681, characteristic = 2 and mantissa = 0.4681

Step 2: Use the antilog table to find the base value.
The row number can be identified using the first two digits in the mantissa and the column from the third digit, and then from the antilog table, find the corresponding number.

  
Here, 0.46 is the row number and 8 is the column number.  

The number is 2936

Step 3: Apply the mean difference
Use the fourth digit of the mantissa to find the mean difference from the same row. Add it to the value we get from step 2. 

Here, the fourth digit of the mantissa is 1

 Add it to 2936 to get 2937.

Step 4: Placing the decimal point
Place the decimal point so that the total number of digits before the decimal equals characteristic + 1.

Here, it is 293.7
The antilog of 2.4681 is 293.7

Let’s find the antilog of a number without the antilog table. We use the formula antilog(x) = 10^x, where x is an integer, to find the antilog.

Let’s find the antilog(2.345)

Using the formula: antilog(x) = 10^x

Antilog of 2.345 = (2.345) = 10^2.345

Breaking down 102.345: \(102.345 = 102 \times 10^{0.345} \)

We use a calculator to find the value of 102 and 10^0.345
102 = 100
10^0.345 ≈ 2.219

Multiplying the parts: 100 × 2.219 = 221.9 

So, antilog(2.345) = 221.9 
 

Antilog tables are used to simplify the calculations involving multiplication, division, exponents, and roots. Follow these steps to find the antilog:

Step 1: Take the log of the number or expression

Step 2: To expand the logarithmic expression, use the properties of logarithms:

log(xy) = log x + log y
log(x/y) = log x - log y
log xm = m log x

Step 3: Find the logarithm of numbers using the log table and combine the results into a single number

Step 4: Use the antilog table to find the antilog of the logarithmic value we get in step 3.
 

The antilog table is a tool we use in science, engineering, and finance to find the number when it is given in logarithmic form. Here are some of the real-life applications of the antilog table. 

Science and Chemistry: Antilog tables are used to calculate pH values, concentrations, and reaction rates. They help to convert the logarithmic measurements back into the actual quantities for experiments and chemical analysis.

Physics and Engineering: Antilog calculations are used in exponential growth and decay, such as radioactive decay, capacitor discharge, and sound intensity. Engineers rely on these values for accurate measurements and design calculations.

Computer Science: Antilog computations support algorithms with logarithmic relationships, data compression, and information scaling. They allow software to convert log-based values into actual quantities.

Robotics: In robotics, an antilog tables are used in sensor data interpretation, motor control, and navigation algorithms. Robots often rely on the logarithmic sensor readings like distance, light, or force, and antilogs convert these readings into real-world measurements for accurate movement and decision-making.

Sound Processing: Antilog calculations help in decibel scaling, audio amplification, and frequency analysis. Sound intensity and volume often use logarithmic scales, and antilogs convert these values back into actual sound pressure levels for playback, recording, and acoustic design.