Antilog Table
Antilog table is used to find the antilogarithm of a number. Antilog is a function that is the inverse of the log function. We know that we use the log table for doing math calculations easily without using a calculator. While doing the calculations, we apply the log first for the given expression, and after simplifying we should use the antilog table to find the antilog of the result that gives the simplified result of the given expression.
Let us learn more about the antilog table along with how it looks like and how to use it for positive and negative numbers. We will solve examples using the antilog table for a better understanding of its usage.
1.  What is Antilog Table? 
2.  How to Use Antilog Table? 
3.  Antilog of a Number Without Using Anti log Table 
4.  Antilog Table in Calculations 
5.  FAQs on Antilog Table 
What is Antilog Table?
Antilog table gives the antilog of a positive or a negative number. Antilog is the inverse of the logarithmic function. i.e., if log x = y then x = antilog (y). i.e., if "log" moves from one side to the other side of the equation, it becomes an antilog. So
log x = y ⇒ x = antilog (y) .... (1)
But by using the log formula, we can convert a logarithmic equation into an exponential equation. From this,
log x = y ⇒ x = 10^{y} ... (2)
From (1) and (2), we can say that antilog(y) = 10^{y}. This can be referred to as the antilog formula. For example:
 antilog (2) = 10^{2} = 100
 antilog (3) = 10^{3} = 0.001
 antilog (3.572) = 10^{3.572} = ?
We can find the last antilog only using the calculator to be antilog (3.572) = 10^{3.572} ≈ 3732. But how can we find this antilog without using a calculator? The solution is the antilog table. Here is the antilog table for the common logarithm.
The antilogarithm table is divided into 3 blocks.
 The first block is the first column (main column) has numbers from .00 to .99.
 The second block (differences columns) shows the digits from 0 to 9.
 The third block (mean differences columns) shows the digits from 1 to 9.
How to Use Antilog Table?
Antilog is always calculated for the logarithm of some number. We know that the logarithm of any number can be expressed as characteristic + mantissa where the characteristic can be either positive or negative whereas the mantissa should always be positive. Let us just recall how to separate the characteristic and mantissa from the logarithm of a number. Be careful while finding them in case of negative numbers (in case of negative numbers we add and subtract 1 to make mantissa positive).
Log of Number  Characteristic + Mantissa  Characteristic  Mantissa 

3.5723  3 + 0.5723  3  0.5723 
0.0052  0 + 0.0052  0  0.0052 
3.278  3  0.278 = (3  1) + (1  0.278) = 4 + 0.722 
4  0.722 
1.27  1  0.27 = (1  1) + (1  0.27) = 2 + 0.73 
2  0.73 
How to Find Antilog?
Here is the process of finding the antilog of a number using the antilog table. Let us assume that we are going to find antilog (3.5723) (which is the first example of the above table).
 Step  1: Find the characteristic and mantissa.
Here, characteristic = 3 and mantissa = 0.5723.  Step  2: Concentrate only on mantissa in this step. Use the first two digits after the decimal point to be the row number and the third digit to be the column number and find the corresponding number from the log table.
The number that lies in row 0.57 and column 2 is 3733.  Step  3: In the same row, look for the mean difference corresponding to the 4th^{ }digit of mantissa. Add this to the value from Step  2.
Here, the 4^{th} digit of the mantissa is 3 and the corresponding mean difference is 3.
3733+3 = 3736  Step  4: Put a decimal point right after the first digit (of the number from Step  3) always.
Then it becomes 3.736.  Step  5: Multiply the number from Step  4 by 10^{characteristic} and the result itself is the antilog of the given number.
Antilog (3.5723) = 3.736 × 10^{3} = 3736.
Antilog of a Number Without Using Anti log Table
In the first section, we have seen the antilog formula to be antilog(x) = 10^{x}. But this formula can be used without a calculator only when x is an integer. If x is NOT an integer, we will have to use a calculator to compute 10^{x}. Let us crosscheck the above antilog using this formula.
Antilog (3.5723) = 10^{3.5723 }≈ 3735, which is very close to the earlier answer, and hence our antilog was correct. Here are more examples:
 Antilog of 1 = 10^{1} = 10
 Antilog of 2 = 10^{2} = 100
 Antilog of 3 = 10^{3} = 1000
 Antilog of 4 = 10^{4} = 10000
Antilog Table in Calculations
The main purpose of the log and antilog tables is to make the process of doing multiplication, division, finding exponents, and roots easier. For simplifying any expression involving product, quotient, or exponents:
 Apply log first.
 Use the following properties of logarithms to expand the log.
log (xy) = log x + log y
log (x/y) = log x  log y
log x^{m} = m log x  Use the log table to find the logarithm of every number and simplify the expression to a single number.
 Find the antilog of the number from the previous step using the antilog table and that will be the final result of the given expression.
Example: Evaluate √(0.00153) using log and antilog table.
Solution:
Let x = √(0.00153) = (0.00153)^{1/2}.
Apply log on both sides:
log x = log (0.00153)^{1/2}
= 1/2 log (0.00153) (Using the property of logarithms)
= (1/2) (2.8153) (Using the log table)
= 1.40765
Now, take antilog on both sides. Then
x = antilog (1.40765)
Here, 1.40765= 1  0.40765 = (1  1) + (1  0.40765) = 2 + 0.59235.
Here, mantissa is 0.59235. So look for the value (in the antilog table) in the row labelled 0.59 and column 2 and add the same row's mean difference under column 3 (we are ignoring the 5th digit which is 5 here as the antilog table can be used only till 4 digits). Then we get 3908 + 3 = 3911.
Now place a decimal point right after the first digit, we get 3.911. Multiply this by 10^{characteristic} = 10^{2}.
Then x = antilog(1.40765) = 3.911 × 10^{2} = 0.03911.
Therefore, √(0.00153) ≈ 0.03911.
We can crosscheck the result using the calculator.
Important Notes on Antilog Table:
 Antilog (x) is the same as 10^{x }and hence the antilog of any number (positive or negative) is always positive.
 When we find the antilog of a number, use its mantissa to see its corresponding number from log table, and use its characteristic for placing the decimal point.
 When we are doing any calculation using logarithms, we always have to apply antilog after simplifying the log of the expression.
 Make sure that mantissa is positive.
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Antilog Table Examples

Example 1: Find the antilog of the following numbers using the antilog table: (a) 0.0052 (b) 3.2778.
Solution:
(a) 0.0052 = 0 + 0.0052.
Its characteristic is 0 and mantissa is 0.0052.
Look for the number (in the anti logarithm table) in row .00 and column 5 and add the corresponding mean difference under 2.
So we get 1012+0 = 1012.
Then antilog (0.0052) = 1.012 × 10^{0 }= 1.012.
(b) 3.2778= 3  0.2778
= (31) + (1  0.2778)
= 4 + 0.7222Its characteristic is 4 and mantissa is 0.7222.
Look for the number in row .72 and column 2 and add the corresponding mean difference of column 2.
Then we get 5272 + 2 = 5274.
Antilog (3.2778) = 5.274 × 10^{4 }= 0.0005274.
Answer: (a) Antilog(0.0052) = 1.012 (b) Antilog (3.2778) = 0.0005274.

Example 2: Verify the answers of Example 1 using the antilog formula and a calculator.
Solution:
By Antilog formula, antilog(x) = 10^{x}
(a) Antilog(0.0052) = 10^{0.0052} = 1.012
(b) Antilog (3.2778) = 10^{3.2778 }= 0.0005274
Both answers are very close to the answers in Example 1.
Answer: The answers are verified using a calculator.

Example 3: Multiply 6.723 × 21.572 using log and antilog tables.
Solution:
Let x = 6.723 × 21.572.
Taking log on both sides,
log x = log (6.723 × 21.572)
Using one of the properties of logarithms,
log x = log 6.723 + log 21.572
Using the log table,
log x = 0.8276 + 1.334
log x = 2.1616Using anti logarithms,
x = antilog (2.1616)
Using antilog table,
x = 145.1Answer: So the approximate value of 6.723 × 21.572 is 145.1.
FAQs on Antilog Table
What is Antilogarithm Table?
The antilogarithm table gives the antilog of a positive or a negative number. Antilog table is used to find the antilogarithm of a number. Antilog is a function that is the inverse of the log function.
Why do We Use Anti logarithm Table?
We use the anti logarithm table to calculate the antilog of a number. For example, if we have a logarithmic equation like log x = y, then we can find x by using x = antilog (y).
What is Antilog 1?
We know that antilog (x) = 10^{x}. Thus, antilog (1) = 10^{1} = 10.
How to Use Antilog Table to Find Antilog of a Number?
Steps to use antilog table to find the antilog of a number:
 Find characteristic and mantissa of the number.
 Use the mantissa to see the corresponding number in the antilog table.
For this, use the first two digits after the decimal point of mantissa to be the row number, the third digit to be the column number, find the corresponding value and add the corresponding mean difference.  Place a decimal point immediately after the first digit of the number from the antilog table.
 Multiply the above number by 10^{characteristic}.
How to Convert Log into Antilog?
Log and antilog are two functions that are inverses of each other. So when log goes to the other side of the equation, it becomes an antilog. For example, if log (m) = n then m = antilog (n).
How to Calculate Antilog With a Calculator?
We don't have an "antilog" button on any calculator. The antilog of a number is equal to 10 raised to the number. For example, antilog (5) = 10^{5}. So we use the formula antilog (x) = 10^{x} to find the antilog using a calculator.
How to Convert Antilog into Log?
As antilog is the inverse of log, whenever antilog (x) = y, it means that x = log y. i.e., if "y is the antilog of x" then "x is the logarithm of y".
What is the Value of Antilog of 2?
We know that the antilog of a number is obtained by raising 10 to that number. Hence, antilog (2) = 10^{2} = 100.
What is Common Anti Logarithm Table from 1 to 10?
Since antilog(x) = 10^{x}, for any x, here is the anti logarithm table from 1 to 10.
Number  Antilog 

1  10^{1} 
2  10^{2} 
3  10^{3} 
4  10^{4} 
5  10^{5} 
6  10^{6} 
7  10^{7} 
8  10^{8} 
9  10^{9} 
10  10^{10} 
Does Antilog Exist for Negative Numbers?
The logarithm of a number can be either positive or negative. Since an antilog is the inverse of log, yes, antilog exists for negative numbers as well. To conclude:
 Log exists only for positive numbers but it results in positive and negative numbers.
 Antilog exists for both positive and negative numbers but it results only in positive numbers.
What is the Difference Between Log and Antilog?
Log  Antilog 

The logarithm of a number x is y (i.e., log (x) = y) if x = 10^{y}.  The antilog of a number x is y (i.e., antilog (x) = y) if y = 10^{x}. 
Log is the inverse function of antilog.  Antilog is the inverse function of log. 
log 10^{x} = x itself.  antilog (x) = 10^{x}. 
Log can be found only for positive numbers.  Antilog can be found only for either positive or negative numbers or for 0. 
Log of a number can be either positive or negative or 0.  Antilog of a number is always positive as it is the result of exponent of 10. 
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