How to Use a Log Table with Formula and Solved Examples
The concept of log table plays a key role in mathematics and is widely used in exams, scientific calculations, and competitive tests. With a log table, students can find logarithmic values without a calculator, speeding up multiplication, division, and exponent-based problems.
What Is Log Table?
A log table is a reference chart or printed table that displays pre-calculated logarithm values for numbers, usually to base 10. It provides a simple way for students to look up log values and their corresponding antilogs for fast and accurate calculations. You’ll find this concept applied in areas such as logarithms, scientific notation, and exam-based calculation shortcuts.
Key Formula for Log Table Usage
Here’s the standard formula used while working with logarithms:
\( \log_{10} N \ = \ \text{Characteristic} + \text{Mantissa} \)
Structure of a Log Table
A typical log table consists of rows and columns. The rows represent the first 2 digits (like 23 in 23.78), and columns represent the next digit. There's also a "mean difference" section for more precise results. The log table only works for up to 4 significant digits. There’s also an antilog table for reverse calculations. Here is a sample format:
| Row (First 2 digits) | Column (Next digit) | Mean Difference |
|---|---|---|
| 23 | 7 | 8 (for 4th digit) |
| 55 | 2 | 0 |
How to Use Log Table: Step-by-Step Illustration
- Write the number in scientific notation (e.g., 23.78 = 2.378 × 101).
- Find the characteristic (the exponent of 10).
If N > 1: Number of digits before decimal - 1 = Characteristic.
If N < 1: -(no. of zeros after decimal + 1) = Characteristic. - For the mantissa, ignore the decimal, pick the first 4 digits.
- In the log table, locate the row for the first two digits, column for the third digit, and add mean difference for the fourth digit.
- Add the characteristic and mantissa.
- The sum gives the logarithm value.
Example: Find log 5520:
1. Scientific notation: 5.520 × 103 (Characteristic: 3 - 1 = 2)
2. Mantissa: Use 5520 → Row 55, Column 2 in log table.
3. Suppose value found is 0.7419.
4. Log(5520) ≈ 2.7419
Solved Examples Using Log Table
Let’s solve a division using log tables:
Calculate \( (17.56 \times 37) / (4.75 \times 24) \) using log table:
2. log 17.56 ≈ 1.2445
log 37 ≈ 1.5682
log 4.75 ≈ 0.6767
log 24 ≈ 1.3802
3. Apply properties:
(log 17.56 + log 37) − (log 4.75 + log 24) = (1.2445 + 1.5682) − (0.6767 + 1.3802) = 2.8127 − 2.0569 = 0.7558
4. Find the antilog of 0.7558 via the antilog table (about 5.70).
5. Final Answer: ≈ 5.70
Key Logarithm Rules (Log Table Essentials)
- log (ab) = log a + log b
- log (a / b) = log a − log b
- log (an) = n × log a
- log 1 = 0; log 10 = 1
Remember these during calculations! These rules help simplify work with log tables in exams and practical questions.
PDF Download & More Resources
You can download a ready log table PDF here for offline use and exam practice. For instant calculations, browse direct log values 1 to 10 for quick reference.
Speed Tip for Exams
Practice quickly identifying the correct row, column, and mean difference in your log table. Pre-memorize common log values (like 0.3010 for log 2, 0.4771 for log 3) to save time in MCQ-based exams or wherever calculators are banned.
Frequent Errors and Misunderstandings
- Not separating the characteristic and mantissa correctly.
- Using the wrong number of digits (always use up to 4 for mantissa).
- Forgetting log tables are only for base 10 (use change of base rule if needed).
- Mistakes in reading mean differences in the column.
Relation to Other Concepts
The idea of log table connects closely with topics like logarithmic functions, logarithmic differentiation, and exponents and powers. Mastering log tables directly boosts speed in those advanced chapters and is vital for calculation-based questions in board exams.
Classroom Tip
A simple way to remember how to use a log table: Think "RCM"—Row (first two digits), Column (third digit), Mean difference (fourth digit). Vedantu’s tutors often share this strategy in live classes for speed and accuracy.
We explored log table: its definition, step-by-step usage, solved examples, exam tips, and how it links to other maths areas. Keep practicing using log tables with Vedantu’s worksheets, calculators, and live guidance to become a confident, error-free solver in both classroom and competitive scenarios.
Related Vedantu Topics
FAQs on Log Table Explained for Easy Calculations
1. What is a log table in mathematics?
A log table is a mathematical table that provides the logarithmic values of numbers to simplify multiplication, division, powers, and roots. Before calculators, log tables were widely used to perform complex calculations quickly.
- Most tables list common logarithms (base 10).
- They usually contain a characteristic and a mantissa.
- They help convert multiplication into addition and division into subtraction.
2. What is the difference between characteristic and mantissa in a log table?
The characteristic is the integer part of a logarithm, while the mantissa is the decimal part found in the log table. For example, if log₁₀(23) = 1.3617:
- 1 is the characteristic.
- 0.3617 is the mantissa.
- The mantissa is always positive in common log tables.
3. How do you use a log table to find the logarithm of a number?
To use a log table, locate the mantissa in the table and then add the correct characteristic based on the number’s digits.
- Step 1: Express the number in standard form (e.g., 23.4).
- Step 2: Use the first two digits (23) to find the row.
- Step 3: Use the third digit (4) to find the column.
- Step 4: Add the characteristic (since 23.4 lies between 10 and 100, characteristic = 1).
4. How do you find the antilog using a log table?
To find an antilog, look up the mantissa in the antilog table and place the decimal point using the characteristic.
- Step 1: Separate characteristic and mantissa (e.g., 2.5463).
- Step 2: Find 0.5463 in the antilog table.
- Step 3: Suppose the table gives 3.51.
- Step 4: Since characteristic is 2, move the decimal 2 places → 351.
5. Why were log tables used before calculators?
Log tables were used because they simplify multiplication and division into addition and subtraction using logarithmic rules. The key properties are:
- log(ab) = log a + log b
- log(a/b) = log a − log b
- log(aⁿ) = n log a
6. How do you multiply numbers using a log table?
To multiply using a log table, add the logarithms of the numbers and then take the antilog of the result.
- Step 1: Find log(12) and log(5).
- Step 2: Add them.
- Step 3: Find the antilog of the sum.
- Example: log(12) ≈ 1.0792, log(5) ≈ 0.6990.
- Sum = 1.7782 → antilog ≈ 60.
7. What is a common logarithm in a log table?
A common logarithm is a logarithm with base 10, written as log₁₀x or simply log x. Most standard log tables are based on base 10.
- Example: log₁₀(100) = 2.
- Example: log₁₀(1000) = 3.
- Used widely in school mathematics and practical calculations.
8. How do you find the square root using a log table?
To find a square root using a log table, divide the logarithm of the number by 2 and then take the antilog.
- Step 1: Find log(49) = 1.6902.
- Step 2: Divide by 2 → 0.8451.
- Step 3: Take antilog → 7.
9. What are the parts of a standard log table?
A standard log table consists of rows, columns, mean differences, and mantissa values. Its main parts include:
- Row numbers (first two digits of the number)
- Column numbers (third digit)
- Mean difference columns (for interpolation)
- Mantissa values (decimal part of log)
10. What are common mistakes when using a log table?
Common mistakes when using a log table include incorrect characteristic placement and wrong decimal adjustment in antilog. Frequent errors are:
- Forgetting to determine the correct characteristic.
- Misreading row or column values.
- Ignoring mean differences for accuracy.
- Placing the decimal point incorrectly after finding antilog.
