Log Table
2026-02-28 10:49 Diff
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Last updated on December 15, 2025

Without a calculator, large calculations, multiplication, division, squares, and roots can be done using a log table. Common logarithms (base 10) show the exponent needed to reach a number. Log tables help find values like log 125 approximately.

What is the Log Table?

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A logarithm table is a mathematical reference used to determine the logarithm of a number with respect to a specific base. Separate log tables exist for different bases such as 10, e (Euler’s number), and 2. Among these, the base-10 logarithm called the common logarithm is the most frequently used and is denoted by log or log₁₀.


A standard common logarithm (base-10) table is arranged into multiple columns for easy reference:
 

  • The main column contains two-digit numbers from 10 to 99, representing the first two digits of the number.
     
  • The next column displays logarithmic values corresponding to the third digit (0–9).
     
  • The mean difference column provides minor corrections to the fourth digit to improve accuracy.

Difference Between Log and Antilog Table

Log and antilog tables are mathematical reference tools used to simplify complex calculations. The difference between the log and antilog tables is: 

Log Table Antilog Table A log table helps convert numbers into their logarithmic form. An antilog table performs the inverse operation, converting logarithmic values back into numbers. A logarithm table is typically based on either base 10 or base e. An antilogarithm table is typically based on either base 10 or base e. Contains logarithmic values of numbers. Contains antilogarithmic (exponential) values of logarithms. Helpful in simplifying multiplication and division. Helpful in solving exponentiation and power-related calculations. Example: log10(2) = 0.3010 Example: antilog10(0.3010) = 2

How to Use a Log Table?

To use a logarithm table (base 10) to calculate the logarithm of a number, we must understand that a logarithm consists of two components: characteristic and mantissa, and both are separated by a decimal point.

For instance, log10 23.78 = 1.3762, which can be found by using a scientific calculator or a common logarithmic table (base 10). In this value, 1 is the characteristic (the integer part) and 0.3762 is the mantissa (the fractional part) of the given number. 

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What are the Characteristics of a Logarithm of a Number?

The characteristic of a logarithm is the whole number part of the logarithm. It represents the exponent of 10 when the number is written in scientific notation. Depending on the number, the characteristic can be positive, zero, or negative.

The characteristic does not depend on the log table. It can be found easily by counting digits using simple rules:

  • If the number is greater than 1: Characteristic = (number of digits to the left of the decimal) − 1.
  • If the number is less than 1:

    Characteristic = −(number of zeros after the decimal before the first non-zero digit + 1).

For example:

For 23.78:

Two digits are to the left of the decimal. Characteristic \(= 2 − 1 = 1.\)

For 0.172:

No zeros after the decimal before the first non-zero digit. Characteristic \(= −(0 + 1) = −1\).

What is Mantissa (Only Positive)?

A log table, which is always a positive number prefixed by a decimal point, can be used to determine the mantissa of the logarithm of a number. Now, let us try to figure out the mantissa of the log10 (0.001724). 

Step 1: Find the first non-zero digit of the given number. 
Here, the first non-zero digit is 1. 

Step 2: Ignore the decimal point and take the next four digits from the first non-zero digit. 
Hence, we have 1724. 

Step 3: The log table’s row number is represented by the first two digits (17) in the number 1724, while the column number is represented by the third digit (2).

Therefore, find the value from the table where this row and column intersect. Here, the matching number is 2355. 

Step 4: If the number has only three digits, the table value is directly the mantissa.

Example: To find log₁₀(172), look up 172 in the table. The value 0.2355 is the mantissa. No mean difference is needed.

If the number has a fourth digit, use the mean difference from the table corresponding to that digit.

Example: For 1724, the fourth digit is 4. Find the mean difference for 4 in row 17 and add it to the table value to get the mantissa. Here, 10 is the corresponding mean difference. 

Step 5: After that, sum the two numbers from step 3 and step 4. 

\(2355 + 10 = 2365 \)

Step 6: Place a decimal point in front of the number to get the mantissa. 

Hence, the mantissa of log 0.001724 is 0.2365. 
Since 0.001724 < 1, we calculate the characteristic. 

The formula is: 

Characteristic = -(number of zeros after the decimal point before a non-zero digit + 1) 
                      \( = -(2 + 1) = -3\)

Logarithm = characteristic + mantissa 
                \( = -3 + 0.2365 = -2.763\)

Hence, log10 (0.001724) ≈ -2.763.

Keep in mind that we can only find the mantissa if the number from step 2 has four or fewer digits. If the number is less than 4 digits, like 18, treat it as 1800 and determine the mantissa.  

How to Find the Logarithm of a Number by Log Table?

We must follow certain steps to use a logarithm table to determine a number’s logarithm. 

Step 1: Find the characteristic of the number, which is the integer part of the logarithm. 

Step 2: Find the mantissa using a log table. 

Step 3: Add the characteristic and mantissa together. 

To understand this, let us look at an example. For instance, the given number is 23.78.

Step 1: We apply the following formula, since the given number is greater than 1:

Characteristic = (number of digits to the left of the decimal point - 1)
Here, 2 digits precede the decimal point in 23.78. 
Hence, characteristic \(= 2 - 1 = 1 \)

Step 2: Since 23.78 is the given number, ignore the decimal point and take it as 2378. 
The first two digits = 23 
Thus, the log table’s row number is 23. 

The third digit = 7 
Hence, 7 is the column number. 
Next, determine where row 23 and column 7 intersect. 
Therefore, the corresponding number is 3747.

Now, the fourth digit = 8 
Find the mean difference for 8 in row 23. 
Therefore, the mean difference is 15. 

Then, add the mean difference of 15 and the number, 3747. 

\(3747 + 15 = 3762  \)

Place a decimal point for the result. 
Mantissa = 0.3762 

Step 3: To determine the logarithm of 23.78, add the characteristic and mantissa. 

\(1 + 0.3762 = 1.3762 \)
\(log (23.78) = 1.3762\)

How to Use a Log Table in Calculations?

To perform complex calculations like multiplication, division, and exponents, we can use logarithms, especially without a calculator.

The properties of logarithms are as follows:

  • \(\log(mn) = \log m + \log n\)
  • \( \log\left(\frac{m}{n}\right) = \log m - \log n \)
  • \(\log(m^n) = n \log m\) 

Let us now examine how to use a log table in calculations using an example. Find (17.56 × 37) / (4.75 × 24) by applying the logarithm table approach. 

Step 1: Determine the logarithm of the given expression using the logarithm properties. \( \log\left(\frac{m}{n}\right) = \log m - \log n \)

\((17.56 × 37) / (4.75 × 24)  \)

\(= log (17.56 × 37) - log (4.75 × 24) \)

\(= log (17.56) + 7 log (3) - (log (4.75) + 4 log (2)) \)

\(= log (17.56) + 7 log (3) - log (4.75) - 4 log (2)\)

We can now determine each logarithm’s characteristic and mantissa using the log table.

x

Characteristic

Mantissa

log x
(characteristic + mantissa)

17.56

1

0.2445 1.2445  

3

0 0.4771 0.4771

4.75

0 0.6767 0.6767

2

0 0.301 0.301

Next, add the values:

  \( = 1.2445 + 7 (0.4771) - 0.6767 - 4 (0.301)\)

\(   = 1.2445 + 3.3397 - 0.6767 - 1.2040\)

\(   = 4.5842 - 1.8807  \)

 \(  = 2.7035\)

Step 2: Next, use the anti-log table to determine the above number’s antilogarithm. 

Antilog (2.7035), which is equal to 102.7035

\(10^{2.7035} = 102 × 10^{0.7035}  \)

              \(= 100 × Antilog (0.7035) \)

Now, estimate Antilog (0.7035):

Antilog (0.7035)  5.057

So, \(10^{2.7035} ≈ 100 × 5.057 = 505. 7 \)

Thus,  (17.56 × 37) / (4.75 × 24) is approximately equal to 505.7

How to Use a Log Table for Natural Logarithms?

To determine natural logarithms (logarithms with base e, where e ≈ 2.718), we use natural log tables, often labeled as ln tables. These tables list the values of ln(x), where x is a positive real number. Unlike common log tables, natural log tables are specifically designed to help find logarithms to base e.  

Using a change of base rule, e = 2.718 can be written as follows:


    \( (log x) / (log e) = (log x) / (log 2.718) \)

Next, use the logarithm table to determine the values of log x and log 2.718 separately, then divide the results.  

For example, determine the value of ln 10. 


log10 e ≈ 0.4343, this is the common logarithm of Euler’s number 


e ≈ 2.718 

\(\ln 10 = \frac{\log_{10} x}{\log_{10} e} = \frac{\log_{10} 10}{\log_{10} 2.718}\)

       \(  = (1.0000) / (0.4343) \)

Hence, ln \(10 =  1.0000 / 0.4343 = 2.3026 \)

Logarithmic Table 1 To 10

A common logarithm table lists logarithms with base 10, also known as decimal logarithms. These values show the power to which 10 must be raised to obtain a given number. The log values for the numbers from 1 to 10 are: 

Number(x) log10(x) 1 0 2 0.3010 3 0.4771 4 0.6020 5 0.6989 6 0.7781 7 0.8450 8 0.9030 9 0.9542 10 1

Tips and Tricks to Master Log Table

Log tables can seem challenging for children, but with parental guidance, they become manageable. By understanding basics, practicing step-by-step, and using simple tips, parents can help their child build confidence, accuracy, and a positive attitude toward logarithms.

  • Before your child uses a log table, make sure they understand what logarithms represent, that it’s the power to which a number must be raised to get another number. Encourage them to see it as the “reverse” of exponentiation.
     
  • Start with easy numbers like 2, 3, 5, 10 to practice finding logs. Once confident, move to more complex numbers. Early success boosts confidence.
     
  • Make a small reference sheet with steps and tips for using the log table. This boosts independence and reduces errors.
     
  • Some numbers have patterns in logs (like powers of 10 or multiples of 2). Recognizing patterns helps your child estimate and cross-check answers quickly.
     
  • Parents can help children estimate approximate answers mentally before checking in the log table. This improves number sense and reduces dependency.
     
  • Teachers can explain that a log x calculator or a logarithm calculator can be used to verify the answer. 
     
  • Parents may provide a simple logarithm book to help students become comfortable reading tables rather than relying solely on electronic devices.
     
  • Teachers can provide structured worksheets that focus on reading mantissas, characteristics, and mean differences, reinforcing correct table usage.

Common Mistakes and How to Avoid Them on Log Table

Without a calculator, complex calculations like multiplication, division, and exponents can be solved using logarithm tables. However, students make mistakes when they work with log tables. Here are some common errors and their helpful solutions to avoid confusion and incorrect answers. 

Real-Life Applications of Log Table

Understanding how log tables work will help students apply them in various situations and fields. The practical uses of the log table are listed below: 
 

  • Science: In science and chemistry, log tables are used to measure acidity or alkalinity, calculate radioactive decay, and solve problems that involve very small or very large quantities. They help scientists and students perform accurate calculations quickly without relying solely on calculators.
     
  • Engineering: Engineers use log tables to manage calculations involving sound intensity, signal strength, and electrical circuits. Before calculators, log tables were essential for designing machines, amplifiers, and communication systems. They help convert complicated multiplication and division into simpler steps.
     
  • Space Science: Log tables are important in fields like geology, seismology, and astronomy. They help scientists measure earthquake magnitudes, compare star brightness, and calculate distances in space. Using log tables makes it easier to handle extremely large numbers efficiently.
     
  • Computer Science and Information: Even in computer science, log tables were historically used to analyze algorithms and solve problems involving large datasets. They help programmers understand how numbers grow and how processes scale efficiently.
     
  • Finance and Business: Log tables are also useful in finance and business for understanding growth, investment calculations, and comparing large numbers. They allow analysts and investors to perform calculations faster and understand trends more clearly.

Problem 1

Find the value of log₁₀(37.28)

Okay, lets begin

1.5730.

Explanation

Step 1: Take the first four digits, 3728.

Step 2: In the log table, find the row for 37 and the column for 2.

You get 0.5717.

Step 3: The fourth digit is 8. The mean difference for 8 = 13.

Add it: 5717 + 13 = 5730.

Step 4: Characteristic = (number of digits before the decimal – 1) = 1.

\(log10(37.28) = 1 + 0.5730 = 1.5730\)

Well explained 👍

Problem 2

Find log₁₀(0.00452)

Okay, lets begin

\(\log_{10}(0.00452) = -2.3447\).

Explanation

Step 1: First non-zero digits → 452.

Step 2: Row = 45, column = 2 → table value = 0.6553.

Step 3: Count zeros after the decimal before the first non-zero digit = 2.

Characteristic\(= -(2 + 1) = -3\)

\(\log_{10}(0.00452) = -3 + 0.6553 = -2.3447\)

Well explained 👍

Problem 3

Find (15.2 × 42) / (3.6 × 28) using the logarithm table.

Okay, lets begin

 6.333. 

Explanation

The logarithm of the given expression can be found using the logarithm properties. 

log (m / n) = log m - log n

\(\log\frac{15.2 \times 42}{3.6 \times 28} = \log 15.2 + \log 42 - \log 3.6 - \log 28\)

Next, use the log table to determine the log value. 

x

Characteristic Mantissa log x
(characteristic + mantissa)
 

15.2

1

0.1818

1.1818

42

1

0.6232

1.6232

3.6

0

0.5563

0.5563

28

1

0.4472

1.4472

Now, we can substitute the values:

\(= (1.1818 + 1.6232) - (0.5563 + 1.4472) \)

\(= 2.8050 - 2.0035 = 0.8015.\)

Next, use the anti-log table to determine the antilog of 0.8015. 

Antilog (0.8015) = 6.333 

Thus, \(\frac{15.2 \times 42}{3.6 \times 28} = 6.333\)

Well explained 👍

Problem 4

Multiply 23.7 × 0.452 Using Logs

Okay, lets begin

3. 7372. 

Explanation

Find log₁₀(23.7) → Characteristic 1 + Mantissa 0.3747 → 1.3747

Find log₁₀(0.452) → Characteristic –1 + Mantissa 0.6553 → –0.3447

Add logs → \(1.3747 + (–0.3447) = 1.0300\)

Find antilog → \(10^{1.0300} = 10 \times 10^{0.0300} \approx 10 \times 1.074 = 10.74\).

Well explained 👍

Problem 5

Find (12.6)² using logs

Okay, lets begin

2.162.

Explanation

We use:

log (mn) = n log m

\(\log_{10}(12.6) = 1.1004\)

So, \(\log_{10}\big((12.6)^2\big) = 2 \times 1.1004 = 2.2008\)

Antilog of \(2.2008 = 10^{0.2008} \times 10^2 = 1.59 \times 100\)

\((12.6)^2 \approx 159\).

Well explained 👍

FAQs on Log Table

1.What do you mean by a log table?

Before the advent of calculators and digital tools, mathematicians used log tables to solve complex mathematical problems. It is a mathematical chart to find the logarithms of numbers with the log base 10. 

2.How to calculate the logarithm of a given number using a log table?

Step 1: Find the characteristic 
Step 2: Find the mantissa
Step 3: Add the characteristic and mantissa together to get the logarithm. 
 

3.How to find the characteristic of a number?

If the number is greater than 1, the formula for finding the characteristic is:
     Characteristic = the number of digits on the left side of the decimal point - 1
If the number is smaller than 1, the formula is:
    Characteristic = - (the number of zeros immediately followed by the decimal point + 1 
 

4.How to calculate the mantissa using a log table?

Step 1: Ignore the decimal point. 
Step 2: Use the first 4 significant digits of the number.
Step 3: The first two digits give the row number, and the third digit gives the column number. Write the table value where the row and the column intersect in the log table. 
Step 4: The fourth digit is the mean difference, then add the mean difference to the value we get from step 3. 

5.Can the mantissa be a negative value?

No, the mantissa is a decimal (fractional) part of a logarithm and is always positive. It can be found using the log table, which only gives positive decimal values. It remains positive even if the characteristic is negative. 

6.How can I check if my child is using the log table correctly?

Ask them to solve simple multiplication or division problems using the log table and cross-check the answer manually or with a calculator. Ensure each step, reading the number, finding the log, adding/subtracting, then taking antilog, is correct.

7.Should parents intervene if child struggles?

Yes, but gently. Ask guiding questions, break problems into manageable steps, and encourage them to reason instead of giving direct answers.

Jaskaran Singh Saluja

About the Author

Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.

Fun Fact

: He loves to play the quiz with kids through algebra to make kids love it.