Log Table

We use a logarithm table to find the logarithm of a number for a given base. Each base, such as 10, e (Euler’s number), or 2, has its own log table. The common logarithm (base 10) is the most widely used and is written as log or log₁₀.

A typical base-10 log table has several columns:

 

1. The main column lists two-digit numbers (10–99) representing the first two digits of the number.
   
    
2. The second column shows values for the third digit (0–9).
   
    
3. The mean difference column provides small corrections for the fourth digit to make the result more accurate.

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. 

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 \(= −(0 + 1) = −1\).

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.  
 

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. 

 

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

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.

 

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
   

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 \(10 =  1.0000 / 0.4343 = 2.3026 \)

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.

 

1. 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.
   
    
2. Start with easy numbers like 2, 3, 5, 10 to practice finding logs. Once confident, move to more complex numbers. Early success boosts confidence.
   
    
3. Make a small reference sheet with steps and tips for using the log table. This boosts independence and reduces errors.
   
    
4. 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.
   
    
5. Parents can help children estimate approximate answers mentally before checking in the log table. This improves number sense and reduces dependency.

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.