Grade 11 Indices and Logarithms II: Short Notes

11 ශ්‍රේණිය · 1 වාරය · 3 පරිච්ඡේදය · දර්ශක හා ලඝුගණක - II

90%

of Grade 11 term tests include it (55 of 61)

546

marks in past papers (57 questions)

12 min

full lesson with quick checks

🎯 What you'll learn

Find the logarithm of a number less than 1 in bar form
Find the antilog of a bar logarithm
Add, subtract, multiply and divide bar logarithms correctly
Simplify products, quotients, powers and roots with log tables
Use a scientific calculator for logs, antilogs and trigonometric ratios

No calculator, just a table

A tinsmith in Kurunegala melts a metal block into a ball and wants its radius: r = ∛(3 × 0.113 ÷ (4 × 3.142)) metres. In the O/L exam hall you get no calculator, only a log table. Logarithms turn ×, ÷, powers and roots into simple +, −, × and ÷, even for numbers smaller than 1.

The laws you already know

lg means log to base 10.• lg(ab) = lg a + lg b• lg(a ÷ b) = lg a − lg b• lg aⁿ = n lg a• lg √a = ½ lg a, lg ∛a = ⅓ lg aSo multiplying becomes adding, dividing becomes subtracting, and a root becomes dividing the log.

Logs of numbers less than 1

Write the number in scientific notation first:0.0347 = 3.47 × 10⁻²The power gives the characteristic: −2, written 2̄ (bar two).The table gives the mantissa for 3.47: 0.5403.So lg 0.0347 = 2̄.5403, meaning −2 + 0.5403.The bar sits only on the characteristic; the mantissa is always positive.

🧮 Characteristic from the power of 10

Use for any number: n is the characteristic (a bar if negative), lg a from the table is the mantissa. 0.582 = 5.82 × 10⁻¹, so lg 0.582 = 1̄.7649.

lg(a × 10ⁿ) = n + lg a, 1 ≤ a < 10

Going back: the antilog

To find the number from its log, use the antilog table on the mantissa, then let the characteristic place the decimal point.Antilog of 2̄.5403: mantissa 0.5403 gives 3.470; characteristic 2̄ means × 10⁻².So the number is 0.03470.Quick rule: n̄ gives (n − 1) zeros straight after the decimal point.

Adding and subtracting bar logs

Work on the mantissa first (it is positive), then the characteristics, carrying as usual.• 2̄.5403 + 1̄.8129: 0.5403 + 0.8129 = 1.3532, carry 1. Characteristic: −2 − 1 + 1 = −2. Answer 2̄.3532.• 1̄.2041 − 0.6990: borrow 1: 1.2041 − 0.6990 = 0.5051. Characteristic: −1 − 1 − 0 = −2. Answer 2̄.5051.

✏️ A product with a small number

Use logarithm tables to find the value of 0.0347 × 5.82.

1lg x = lg 0.0347 + lg 5.82
2= 2̄.5403 + 0.7649
30.5403 + 0.7649 = 1.3052: carry 1, so −2 + 1 = −1. lg x = 1̄.3052
4Antilog: 0.3052 → 2.019, and 1̄ means × 10⁻¹. x = 0.2019

✅ x ≈ 0.2019

⚠️ 1̄.8513 is not 1.8513

A very common slip is dropping the bar and adding 1̄.8513 as if it were 1.8513. The answer then comes out 100 times too big. Remember 1̄.8513 = −1 + 0.8513. Sanity-check the size at the end: 0.0347 × 5.82 must be about 0.03 × 6 ≈ 0.2.

Powers and roots of bar logs

Multiply both parts, then carry: 3 × 1̄.8293 = 3̄ + 2.4879 = 1̄.4879.Divide: first make the characteristic exactly divisible, adding the same to the mantissa.½ × 1̄.6992: write 2̄ + 1.6992, halve: 1̄.8496.⅓ × 1̄.7630: write 3̄ + 2.7630, divide by 3: 1̄.9210.

✏️ The tinsmith's ball

Find r = ∛(0.339 ÷ 12.57) using logarithm tables. (This is the radius in metres of a ball of volume 0.113 m³.)

1lg r = ⅓ (lg 0.339 − lg 12.57) = ⅓ (1̄.5302 − 1.0993)
20.5302 − 0.0993 = 0.4309; characteristic −1 − 1 = −2. So 2̄.4309.
3⅓ × (3̄ + 1.4309) = 1̄.4770
4Antilog: 0.4770 → 2.999; × 10⁻¹ gives 0.2999.

✅ r ≈ 0.300 m (30 cm)

⚠️ Forgetting the root

In many answer scripts the log is never divided by 3 for a cube root (or by 2 for a square root), so the 'radius' comes out huge. Write the ⅓ or ½ in front of the bracket on your very first line, and don't take the antilog until you have divided.

Using a scientific calculator

• log gives lg directly: log 0.0347 = −1.4597. That is the same as 2̄.5403, because −2 + 0.5403 = −1.4597.• 10ˣ gives the antilog.• xʸ, √, ∛ for powers and roots.• For sin, cos, tan, check the screen shows D (degree mode) first.

📌 Remember

• For a number below 1, write a × 10⁻ⁿ: lg = n̄ . (mantissa of a).• Only the characteristic is negative; the mantissa stays positive.• Add/subtract mantissas first, then characteristics with the carry.• To divide a bar log, make the characteristic divisible first.• Calculator: log, 10ˣ, and degree mode for sin, cos, tan.

🧾 Quick revision sheet

lg(ab)	lg a + lg b
lg ∛a	⅓ lg a
What does 2̄.5403 mean?	−2 + 0.5403 (= −1.4597)
Characteristic of lg 0.0049	3̄, since 0.0049 = 4.9 × 10⁻³
Antilog of 1̄.3010	0.2 (2.000 × 10⁻¹)
½ × 1̄.6992	Write 2̄ + 1.6992, halve: 1̄.8496
Calculator key for antilog	10ˣ (usually SHIFT then log)
Before sin/cos/tan on a calculator…	make sure it is in degree mode (D).