Using Log Tables · Hook

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Ever wondered how people multiplied big numbers before calculators?

Engineers building roads and bridges in the 1960s used four-figure log tables — the same tables at the back of your WAEC maths set. They turned hard multiplications into easy additions. Let's learn their trick.

Revision notes

📒 Using Log Tables: lesson notes

Read a log as characteristic + mantissa, handle bar notation, and multiply or divide big numbers by adding or subtracting logs.

Engineers building roads and bridges in the 1960s used four-figure log tables — the same tables at the back of your WAEC maths set. They turned hard multiplications into easy additions. Let's learn their trick.

Characteristic and mantissa

Every positive number can be written in standard form, A × 10ⁿ, with A between 1 and 10. For example, 345.6 = 3.456 × 102.

So log 345.6 = log 3.456 + 2. The whole-number part, 2, is the characteristic. The decimal part, log 3.456 = 0.5386, is the mantissa — that is what the table gives.

Quick rule: for a number ≥ 1, the characteristic is one less than the number of digits before the decimal point.

🔑 Key idea: log N = characteristic (power of 10) + mantissa (from the tables).

Numbers less than 1: bar notation

0.03456 = 3.456 × 10−2, so log 0.03456 = −2 + 0.5386.

We keep the mantissa positive (as in the tables) and write the −2 as a bar on top: 2̄.5386, read 'bar two point five three eight six'. The bar belongs only to the characteristic.

The antilog reverses the log: the mantissa gives the digits, and the characteristic tells you where the decimal point goes.

🔑 Key idea: 2̄.5386 means −2 + 0.5386 — only the characteristic is negative.

Calculating with logs

To multiply, add the logs; to divide, subtract them. Then take the antilog of the result.

Example: 23.45 × 4.12 → log 23.45 = 1.3701 and log 4.12 = 0.6149. Adding gives 1.9850. The antilog of 0.9850 is 9.661 and the characteristic 1 means × 10, so the answer is 96.61.

You can also use given values: if log 2 = 0.3010 and log 3 = 0.4771, then log 6 = 0.3010 + 0.4771 = 0.7781.

🔑 Key idea: Multiply → add logs. Divide → subtract logs. Then antilog.

Worked example

Use logarithms to evaluate 902 ÷ 6.23, correct to 4 significant figures. From the tables: log 9.02 = 0.9552, log 6.23 = 0.7945 and antilog 0.1607 = 1.448.

  1. Step 1: Log of each number

    log 902 = 2.9552 (902 = 9.02 × 102)

    log 6.23 = 0.7945 (6.23 = 6.23 × 100)

    Why? The characteristic is the power of 10; the mantissa comes from the tables.

  2. Step 2: Subtract the logs

    2.9552 − 0.7945 = 2.1607

    Why? log (a ÷ b) = log a − log b

  3. Step 3: Antilog

    2.1607 = 2 + 0.1607

    antilog 0.1607 = 1.448, and the characteristic 2 means × 102

    1.448 × 102 = 144.8

Answer: 144.8

Summary

  • Characteristic = the power of 10 in standard form; mantissa = the decimal part from the tables.
  • For a number ≥ 1, the characteristic is one less than the number of digits before the point.
  • Numbers less than 1 have a negative characteristic, written with a bar: 0.0345 → 2̄.5378.
  • Multiply by adding logs, divide by subtracting logs, then take the antilog.

Ready to check you've got it? Scroll up and work through the lesson: the interactive visuals, practice questions and mastery check are all there.