How to Use a Log Table Step-by-Step?
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: Definition, Chart, and Usage in Maths
1. What is a log table in mathematics?
A log table, or logarithm table, is a reference chart containing pre-calculated logarithms of numbers. Its primary purpose is to simplify complex arithmetic calculations involving multiplication, division, exponents, and roots by converting them into simpler operations of addition, subtraction, multiplication, and division, respectively. Most standard log tables use base 10 (common logarithm).
2. What is the main purpose of using a log table in calculations?
The main purpose of a log table is to perform complex calculations quickly and accurately without a calculator. By using logarithm properties, a student can:
- Convert a multiplication problem into an addition problem.
- Convert a division problem into a subtraction problem.
- Convert an exponentiation problem into a multiplication problem.
This makes it an essential tool in exams where electronic calculators are not permitted, as it tests a student's understanding of logarithmic functions.
3. What are the characteristic and mantissa in the context of a log table?
The logarithm of any number consists of two parts: the characteristic and the mantissa.
- The characteristic is the integer part of the logarithm. It is determined by the number of digits to the left of the decimal point in the original number. For numbers less than 1, the characteristic is negative.
- The mantissa is the positive decimal part of the logarithm. This value is found using the log table and depends only on the sequence of digits in the number, not on the position of the decimal point.
4. How do you find the logarithm of a number using a log table?
To find the log of a number, you first separate it into its characteristic and mantissa. For example, to find log(234.5):
- Step 1: Find the Characteristic. The number has 3 digits before the decimal, so the characteristic is (3 - 1) = 2.
- Step 2: Find the Mantissa. Look for the row '23' in the log table. Move across this row to the column '4'. Note this value. Then, in the same row, find the value in the 'mean difference' column for '5'. Add the mean difference value to the main value.
- Step 3: Combine. The final log value is the characteristic plus the mantissa.
5. What is the difference between a log table and an antilog table?
A log table and an antilog table perform opposite functions. A log table is used to find the logarithm of a number (e.g., finding the value of log₁₀(100), which is 2). In contrast, an antilog table is used to find the original number whose logarithm is known (e.g., finding the antilog of 2, which is 100).
6. How are the properties of logarithms applied when using a log table for calculations?
Logarithm properties are essential for using log tables effectively:
- Product Rule (log ab = log a + log b): To multiply two numbers, find their individual logs from the table, add the log values, and then find the antilog of the sum.
- Quotient Rule (log a/b = log a - log b): To divide two numbers, find their individual logs, subtract the second from the first, and then find the antilog of the result.
- Power Rule (log aⁿ = n log a): To find a number raised to a power, find the log of the number, multiply it by the power, and then find the antilog.
7. Why were log tables invented, and are they still relevant in the digital age?
Log tables were invented in the 17th century, primarily by John Napier, to simplify the extremely tedious and error-prone calculations required in fields like astronomy, navigation, and engineering. While calculators and computers have now replaced them for professional work, they remain relevant in education. They help students build a deep, conceptual understanding of how logarithms work and are a crucial skill for exams that restrict calculator use, ensuring students can solve problems from first principles.
8. What are some common mistakes students make when reading a log table?
Common mistakes include:
- Incorrectly calculating the characteristic, especially for numbers less than one (e.g., 0.0234).
- Misreading the table by looking at the wrong row or column.
- Forgetting to add the mean difference value, or adding it incorrectly.
- Confusing the procedure for using the log table with the antilog table.
- Making simple arithmetic errors when adding or subtracting the log values during a larger calculation.
9. How does a log table physically represent the mathematical definition of a logarithm?
A logarithm is defined as the exponent to which a base must be raised to produce a certain number. A log table is a physical manifestation of this definition. The number you look up in the table is the 'result', and the value the table gives you (the mantissa) is the decimal part of the 'exponent'. The characteristic completes this exponent. Therefore, the table serves as a ready-reckoner that maps a number directly to its corresponding power or exponent for a fixed base, usually 10.
10. In which real-world scientific measurements are the principles of logarithms used?
The principles underlying log tables are fundamental to many scientific scales and measurements because they help manage and compare numbers over a very wide range. Key examples include:
- The pH scale in chemistry, which measures acidity and is a logarithmic scale.
- The Richter scale in seismology, used to measure earthquake intensity.
- The decibel scale in physics, which measures sound intensity.
In all these cases, a logarithmic scale makes it easier to represent and work with vastly different quantities.
