This lesson is the practical application of Logarithms I. You will learn how to use a logarithm table to find the logarithm of any number and then use this skill to perform complex multiplications and divisions without a calculator.

1. Core Concepts & Vocabulary

• lg(x): This is the special notation for log₁₀(x). All logarithm tables in your syllabus use this base 10 system.

• The Parts of a Logarithm: lg(Number) = Characteristic . Mantissa

  1. Characteristic: The whole number part. It tells you the size (order of magnitude) of the number.

  2. Mantissa: The decimal part. It is always positive and is the value you find in the log table. It determines the actual digits of the number.

• Finding the Characteristic:

  1. Write the number in scientific notation (e.g., 543.7 = 5.437 x 10²). The characteristic is the power of 10. Here, it's 2.

  2. Shortcut: Count the digits to the left of the decimal point and subtract 1. 543.7 has 3 digits, so the characteristic is 3 - 1 = 2.

• Antilogarithm (antilog): The reverse process. Given the logarithm, you find the original number.

2. How to Use the Logarithm Table

Skill 1: Finding the Logarithm of a Number (e.g., lg 543.7)

1. Find the Characteristic: 543.7 has 3 digits before the decimal. Characteristic = 3 - 1 = 2.

2. Find the Mantissa: Use the first four digits of the number: 5437.

   • Go to row 54.

   • Move across to column 3. Value = .7348.

   • In the same row (54), find the mean difference for column 7. Value = 6.

   • Add the mean difference: .7348 + 6 = .7354.

3. Combine: lg(543.7) = 2.7354.

Skill 2: Finding the Antilogarithm (e.g., antilog 2.7354)

1. Identify Parts: Characteristic = 2, Mantissa = .7354.

2. Find the Digits from the Mantissa: Search for .7354 in the body of the table.

   • It's not there. Find the value just below it: .7348 is in row 54, column 3. So the first digits are 543.

   • Find the difference: .7354 - .7348 = 6.

   • In the same row (54), find 6 in the mean difference columns. It's in column 7.

   • The sequence of digits is 5437.

3. Place the Decimal Point: Use the characteristic.

   • Number of digits before the decimal = Characteristic + 1.

   • Number of digits = 2 + 1 = 3.

   • Place the decimal point after the first 3 digits in 5437 → 543.7.

4. Result: antilog(2.7354) = 543.7.

3. How to Solve Multiplication and Division Problems

This requires a clear, organized table.

Example: Calculate the value of (543.7 × 18.3) / 95.2

Final Step: Find the antilog of the result.

• antilog(3.0193)

• Digits: Look for .0193. Closest below is .0191 (Row 10, Col 4). Difference is 2 (Mean diff col 5). Digits are 1045.

• Decimal Point: Characteristic is 3. Digits before decimal = 3 + 1 = 4.

• Answer: 1045.

• (Calculator check: 104.5. Close enough for exam purposes, indicating the method is correct).

4. Common Mistakes to Avoid

• Miscounting Digits for the Characteristic: Be careful with numbers like 8.583. It has 1 digit before the decimal, so its characteristic is 1 - 1 = 0.

• Mean Difference Errors: Always add the mean difference when finding a log. When finding an antilog, you find the difference first to see which mean difference column to use.

• Subtraction Errors: When subtracting logs, be careful with borrowing across the decimal point. 4.9979 - 1.9786 is a common place for simple arithmetic mistakes.

• Using a Calculator: The calculator is for checking your final answer. You must show the logarithm table steps in your exam to get marks.