Log table (logarithm table) is used in performing bigger calculations (of multiplication, division, squares, and roots) without using a calculator. The logarithm of a number to a given base is the exponent by which that base should be raised to give the original number. For example, if log₂16 = x then 2x = 16 and x = 4 satisfies this equation. So log₂ 16 = 4. But what about log₂ 15? If we assume log₂ 15 = x then we get 2x = 15 and we cannot find the value of x manually here. Log table helps us in finding the value of log₂ 15.
In this article, we will learn how to use a logarithm table. Let us see how to find the logarithm of a number using a log table and how to use logs in performing the calculations along with many more examples.
|1.||What is Log Table?|
|2.||Logarithm Table of Common Logarithms|
|3.||How to Use Log Table?|
|4.||Using Log Table in Calculations|
|5.||How to Use Log Table for Natural Logarithms?|
|6.||FAQs on Log Table|
What is Log Table?
Log table for a given base is a table of logarithms that is used to find the logarithm of a specific number to that particular base. There are different log tables for the bases like 10, e (Euler's number), 2, etc. The logarithm table gives the easiest way to find the value of the log of a number precisely. The logarithmic function is considered as the inverse of the exponential function. We will explore the log table for common logarithms in the next section.
Logarithm Table of Common Logarithms
We know that the logarithm with base 10 is known as a common logarithm and it can be written as log10 (or) just log. Given below is the common log table (i.e., for base 10). It means, it can give the value of log x (which is also written as log10 x) for any x.
The log table mainly has 3 types of columns:
- The first column is called the "main column" and it has numbers from 10 to 99 (all 2 digit numbers).
- The second set of columns is called the "differences column" and it shows the "differences" for the digits 0 to 9.
- The third set of columns is called the "mean differences column" and it shows the mean differences from 1 to 9.
Apart from this table, we have log tables for base e (which is called the natural logarithm table) and for base 2 (which is called the binary logarithm table).
How to Use Log Table?
The logarithm of any number consists of two parts: characteristic and mantissa. These two parts are always separated by a decimal point. For example, log 23.78 = 1.3762, and here, 1 is called the characteristic, and 3762 is called the mantissa. i.e., in the logarithm of a number:
- The integer part (which lies on the left side of the decimal point) is called characteristic;
- The decimal (or) fractional part (which lies on the right side of the decimal point) is called the mantissa.
This scenario will be different if the characteristic is negative. Let us see how to calculate each of these.
Characteristic (Positive or Negative)
The characteristic of the logarithm of a number is the exponent of 10 in its scientific notation. So the characteristic can be either positive or negative. Here are some examples to understand what is characteristic.
|Number||Scientific Notation||Characteristic of Log of Number|
|23.78||2.378 × 101||1|
|4.572||4.572 × 100||0|
|552||5.52 × 102||2|
|0.172||1.72 × 10-1||-1|
|0.0172||1.72 × 10-2||-2|
Thus, the characteristic of logarithm a number doesn't depend on the log table. Here are some helpful tips to calculate the characteristic of the log of a number without actually converting it into scientific notation.
- If the number is greater than 1, then use the formula: characteristic = the number of digits on the left side of the decimal point - 1.
- If the number is less than 1, then use the formula: characteristic = - (the number of zeros immediately followed by the decimal point + 1)
Let us see the same examples (as in the previous table) and calculate their characteristics using these tips.
|If Number > 1|
|Number||Number of digits on the left side of the decimal point||Characteristic|
|23.78||2||2 - 1 = 1|
|4.572||1||1 - 1 = 0|
|552 (552.0)||3||3 - 1 = 2|
|If Number < 1|
|Number||Number of zeros that immediately followed the decimal point||Characteristic|
|0.172||0||- (0 + 1) = -1|
|0.0172||1||- (1 + 1) = -2|
Mantissa (Only Positive)
The mantissa of the logarithm of a number is always positive and is found using the log table. Remember that the mantissa is always prefixed by a decimal point. Here are the steps to find the mantissa of the logarithm of a number. Assume that we are trying to find the mantissa of the logarithm of the number 0.001724.
- Step - 1: Find the first non-zero digit of the number.
In our case, the first non-zero digit is 1.
- Step - 2: Take the next 4 digits from the number obtained in Step - 1 irrespective of the decimal point.
Then we get 1724.
- Step - 3: In the number from Step - 2, the first two digits together give the row number and the third digit gives the column number of the log table. Identify the value from the table that lies at the intersection of this row and column.
The first two digits are 17 and the third digit is 2. So we should see the intersection of the row labeled 17 and column labeled 2.
Then the corresponding number is 2355.
- Step - 4: If there is no 4th digit in the number from Step - 2, then the above number itself is the mantissa. If there is the 4th digit, then we have to look for the mean difference (from the log table) of the same row corresponding to the 4th digit.
The number from Step - 2 is 1724 and it has a 4thdigit to be 4. So look for the mean difference.
So the corresponding mean difference is 10.
- Step - 5: Add both numbers (from Step - 3 and Step - 4).
2355 + 10 = 2365.
- Step - 6: Just prefix the number by the decimal point that gives the mantissa.
The mantissa of log 0.001724 is 0.2365.
Here is the table with a mantissa of the same numbers (as in the previous table).
|Number||Number with 4 digits||Mantissa|
|23.78||2378||3747+15 ⇒ 0. 3762|
|4.572||4572||6599+2 ⇒ 0. 6601|
|552||5520||7419+0 ⇒ 0.7419|
|0.172||1720||2355+0 ⇒ 0. 2355|
|0.0172||1720||2355+0 ⇒ 0.2355|
Note that we can find the mantissa only if the number from Step - 2 is comprised of either 4 digits or less than 4 digits. If a number from Step - 2 has less than 4 digits (say we have 17), then write two zeros next to it (write it as 1700) and then find the mantissa.
Finding Logarithm of a Number by Log Table
Here are the steps to find the logarithm of any number using the logarithm table.
- Step - 1: Find its characteristic (no need for a log table here).
- Step - 2: Find its mantissa (use log table).
- Step - 3: Add both characteristic and mantissa.
Here is a table of examples (see previous tables to see the calculations of characteristic and mantissa) to understand how to find the logarithm of a number using the common logarithm table.
(characteristic + mantissa)
We can cross-check the results obtained in the last column using a calculator as well.
Using Log Table in Calculations
The logarithm of any number is used to do tedious calculations involving multiplication, division, and exponents. For doing these operations, we use the following properties of logarithms.
- log (mn) = log m + log n
- log (m / n) = log m - log n
- log mn = n log m
Let us see how to use the log table in calculations using the following table.
Example: Find (17.56 × 37) / (4.75 × 24) using the logarithm table
The steps to perform this calculation using log table are as follows:
- Step - 1: Find the logarithm of the given expression using the properties of logarithms.
- Step - 2: Find the antilogarithm of the result obtained from Step - 1 and that will be the final result.
Using the properties of logarithms,
log [(17.56 × 37) / (4.75 × 24)]
= log (17.56 × 37) - log (4.75 × 24)
= (log 17.56 + log 37) - (log 4.75 + log 24)
= log 17.56 + 7 log 3 - log 4.75 - 4 log 2
Now using the log table (computing characteristic, mantissa, and adding them),
= 1.2445 + 7 (0.4771) - 0.6767 - 4 (0.3010)
Now, calculate the anti-logarithm of this number using the anti-log table.
Antilog (2.7035) (which is equal to 102.7035) = 505.24.
Therefore, [(17.56 × 37) / (4.75 × 24)] is approximately equal to 505.24.
The answer we got using the logarithm table will be very close to the answer from the calculator. Try to verify.
How to Use Log Table for Natural Logarithms?
In fact, we have a separate set of log tables for natural logarithms. But if we want to calculate the natural logarithm of a number (say, ln x) using the common logarithm tables, then we can follow the procedure below:
- We know that ln x is nothing but loge x. By change of base rule, this can be written as (log x) / (log e) = (log x) / (log 2.718) (because e = 2.718).
- Then find the log x and log 2.718 separately from the common logarithm table and divide the results.
Here is an example.
Example: What is the value of ln 5?
Solution: ln 5 = (log 5) / (log 2.718)
= (0.6990) / (0.4343)
Important Notes on Log Table:
- Log table can be used to find the logarithm of a number if the number of digits after its first non-zero digit is less than or equal to 4. If the number of digits is more than 4, we simply ignore the digits from the 5th digit onwards.
- The log of a number has two parts: characteristic and mantissa and their sum is the logarithm.
- Characteristic can be positive or negative.
- Mantissa should be always positive.
- If we have to find the logarithm of a base (other than 10), then first use the change of base of rule logb a = (log a) / (log b) and then use the same table of common logs.
Examples Using Logarithm Table
Example 1: Find the common log of each of these numbers using the log table. Also, verify the answers using a calculator. (a) 342.2 (b) 0.3422
Observe that both numbers are the same except for the placement of decimal points. So we can find the mantissa just using the 4-digit number 3422. To find the mantissa of 3422, just see the row labeled 34 and column labeled 2, find the number at their intersection in the log table, and add the number which is under mean difference 2 in the same row.
From the log table, 5340 + 3 = 5343
Mantissa of 3422 = 0.5343.
(a) The given number = 342.2 > 1.
So its characteristic = number of digits on the left side of decimal point - 1 = 3 - 1 = 2.
So log (342.2) = characteristic + mantissa = 2 + 0.5343 = 2.5343.
(b) The given number = 0.3422 < 1.
So its characteristic = - (number of zeros on the right side of decimal point + 1) = - (0 + 1) = -1.
So log (0.3422) = characteristic + mantissa = -1+ 0.5343 = -0.4657.
Verify both answers using the calculator using "log" button and we can see we get the same answer.
Answer: (a) 2.5343 (b) -0.4657.
Example 2: The logarithms of some numbers are computed using log table. Identify their characteristics and mantissa. (a) log x = 3.5381 (b) log x = -2.6505.
We know that the logarithm of a number is the sum of characteristic and mantissa. But remember that the mantissa is always positive.
(a) log x = 3.5381
= 3 + 0.5381
= characteristic + mantissa
Thus, characteristic = 3 and mantissa = 0.5381.
(b) log x = -2.6505
= -2 - 0.6505
But mantissa can't be negative. So we add and subtract 1.
= (-2 - 1) + (1 - 0.6505)
= -3 + 0.3495
= characteristic + mantissa
Thus, characteristic = -3 and mantissa = 0.3495.
Answer: (a) characteristic = 3 and mantissa = 0.5381 (b) characteristic = -3 and mantissa = 0.3495.
Example 3: Calculate √282 using the logarithm table and check the answer using the calculator.
Apply log to the given number, simplify, and then apply antilog.
log √282 = log (282)1/2 (By square root formula)
= (1/2) log 282
= (1/2) (2.4502) (using logarithm table)
Now √282 = antilog (1.2251) = 16.79.
Answer: We found that the approximate value of √282 is 16.79.
FAQs on Log Table
How to Compute Logarithm of a Number Using Log Table?
For finding the logarithm of a number using a log table:
- find characteristic
- find mantissa
- just add them both.
How to Find Characteristic and Mantissa From Log Table?
The sum of characteristic and mantissa of a number gives the logarithm of that number.
- For finding the characteristic, don't use the log table. Instead, write the number in scientific notation. Then the exponent of 10 will be the characteristic.
- For finding mantissa, write 4 digits of the given number (starting from the first non-zero digit) ignoring the decimal point. The first two digits of these 4 digits give the row number, the 3rd digit is the column number, and the 4th digit is the mean difference. Just find the number in the log table that is in the intersection point of the row and column and add the mean difference of the 4th digit of the same row to that number. This will give the mantissa.
How to Use Log Table in Calculations?
We can use the log table for doing very difficult calculations involving exponents, multiplication, and division in the easiest way. For doing any calculation:
- Apply log to that calculation.
- Expand it using the properties of logarithms.
- Find the logarithm of each number using the log table.
- Finally, find the antilog of the resultant number.
Do People Still Use Logarithm Table?
Yes, people are still using logarithm tables especially when a calculator is not available (or) not allowed (especially in case of examinations). Using a log table, the calculations in mathematics can be done very faster.
Write Common Log Table From 1 to 10.
Here is the common log (log10 (or) simply log) table from 1 to 10. All answers are rounded to 4 decimals.
Write Natural Log Table From 1 to 10.
Here is the natural log (logₑ (or) simply ln) table from 1 to 10. All results are rounded to 4 decimals.
What is the Use of Logarithm Table?
We can do mathematical calculations with the following operations very easily using the logarithmic table:
Are Log Tables Still Used?
Yes, there are some exams (especially math or physics-related) in some countries up to specific grades where the calculator is not allowed. Then it is very difficult to do some big calculations without a log table.
How to Calculate Logarithm of a 5 digit Number Using Log Table?
We can find mantissa of the logarithm of a number only for 4 digit numbers. For more than 4 digits, we simply ignore the digits after the 4th digit while finding mantissa. But remember that we don't ignore the digits while calculating the characteristic. For example, to find log (1234.5):
- Characteristic = number of digits on the left side - 1 = 4 - 1 = 3.
- Mantissa (see the row labeled 12 and column labeled 3 of log table and add the mean difference of the same row under 4) = 0.0913.
Here, we have ignored the 5th digit (which is 5) while calculating mantissa.
- Add them both. Then log (1234.5) = 3 + 0.0913 = 3.0913. This is not the exact log value, but it is an approximate value, as we have ignored the last digit.