Grade 10 Logarithms II: Short Notes

10 ශ්‍රේණිය · 19 පරිච්ඡේදය · ලඝුගණක II

63%

of Grade 10 term tests include it (34 of 54)

290

marks in past papers (34 questions)

12 min

full lesson with quick checks

🎯 What you'll learn

Find the logarithm of a number less than 1 and write it in bar form
Add, subtract, multiply and divide numbers with bar characteristics
Find antilogarithms with negative characteristics
Use log tables to simplify products, quotients, powers and roots
Use a scientific calculator to find and check logarithms

Tiny numbers, big sums

A goldsmith in Pettah melts a ring into beads of 0.0475 cm³ each, and the term test asks you for ³√0.0475 with only a log table. Numbers smaller than 1 turn up in almost every Paper II volume question. Today you learn to handle them.

Quick recap: two parts of a log

Write the number in scientific notation: 375.2 = 3.752 × 10².• The power of 10 is the characteristic: 2• The table gives the mantissa for 3.752: 0.5743So lg 375.2 = 2.5743.The mantissa depends only on the digits; the characteristic tells you where the decimal point is.

Numbers less than 1: the bar

0.0375 = 3.75 × 10⁻², so the characteristic is −2 and the mantissa is still 0.5740.lg 0.0375 = −2 + 0.5740, written 2̄.5740 ("bar 2 point 5740").The bar sits only on the characteristic. The mantissa stays positive.For 0.375 = 3.75 × 10⁻¹ it is 1̄.5740.

Same digits, same mantissa

The mantissa (blue) never changes. Only the characteristic (amber) changes. Below the red line the numbers are less than 1, so the characteristic carries a bar.

Adding and subtracting bar numbers

Do the mantissas first, then the characteristics, carrying as usual. Bars are negative numbers.Add: 2̄.6 + 1̄.7 → 0.6 + 0.7 = 1.3, carry 1. Then −2 + (−1) + 1 = −2. Answer 2̄.3Subtract: 0.3 − 1̄.8 → 0.3 − 0.8 needs a borrow: 1.3 − 0.8 = 0.5. Then −1 − (−1) = 0. Answer 0.5

⚠️ 2̄.5740 is not −2.5740

O/L examiners report this every year: students mishandle bar characteristics when adding and subtracting. 2̄.5740 means −2 + 0.5740 = −1.4260. Only the 2 is negative. Never put the minus in front of the whole number, and always add the mantissas as positive numbers.

✏️ A product with a small number

Use log tables to find 0.0473 × 6.82.

1lg 0.0473 = 2̄.6749 and lg 6.82 = 0.8338.
2Add: 0.6749 + 0.8338 = 1.5087, carry 1 → −2 + 1 = −1. Total 1̄.5087.
3Antilog of 0.5087 is 3.226.
4Characteristic 1̄ means × 10⁻¹: 0.3226.

✅ 0.3226

Powers and roots of bar numbers

Power: lg 0.7³ = 3 × 1̄.8451. Multiply the mantissa, then the characteristic: 3 × 0.8451 = 2.5353, and 3 × (−1) + 2 = −1. So 1̄.5353.Root: ½ × 1̄.6992. −1 does not halve neatly, so rewrite as 2̄ + 1.6992. Then halve each part: 1̄.8496.

✏️ A cube root

Find ³√0.0475 using log tables.

1lg ³√0.0475 = ⅓ × lg 0.0475 = ⅓ × 2̄.6767.
2−2 is not divisible by 3, so write 2̄.6767 = 3̄ + 1.6767.
3Divide each part by 3: 1̄ + 0.5589 = 1̄.5589.
4Antilog of 0.5589 is 3.622, and 1̄ means × 10⁻¹.

✅ 0.3622

⚠️ Dividing a bar the wrong way

Examiners say many candidates cannot find square or cube roots with logs. A typical slip: writing ⅓ × 2̄.6767 as 1̄.2256, just dividing 0.6767 by 3 and guessing the bar. First make the characteristic a multiple of the divisor, adding the same amount to the mantissa: 2̄.6767 = 3̄ + 1.6767.

Using a scientific calculator

• log key gives lg x: log 0.0375 shows −1.4260.• To get bar form, write −1.4260 = −2 + 0.5740 = 2̄.5740.• 10ˣ (often SHIFT + log) gives the antilog.• Use x², √, ∛ and brackets to work out the whole expression and check your log-table answer. A small difference in the last digit is normal.

📌 Remember

• For a number less than 1, the characteristic is negative: 0.0375 → 2̄.5740.• The bar is only on the characteristic; the mantissa is always positive.• Work the mantissas first, then the characteristics (carry and borrow as usual).• To divide, make the characteristic a multiple of the divisor first.• Check with a calculator: log and 10ˣ.

🧾 Quick revision sheet

What does 2̄.5740 mean?	−2 + 0.5740 (= −1.4260)
Characteristic of lg 0.0049	3̄, because 0.0049 = 4.9 × 10⁻³
Does the mantissa change for 3.75, 0.375 and 0.0375?	No. It is 0.5740 for all; only the characteristic changes.
x = antilog of 1̄.3010	2 × 10⁻¹ = 0.2
lg of a product / quotient	lg(ab) = lg a + lg b; lg(a/b) = lg a − lg b
lg of a power or root	lg aⁿ = n lg a; lg √a = ½ lg a
How do you halve 1̄.6992?	Write 2̄ + 1.6992, then halve: 1̄.8496
Calculator key for an antilog	10ˣ (usually SHIFT + log)