Grade 10 Logarithms I: Short Notes

10 ශ්‍රේණිය · 18 පරිච්ඡේදය · ලඝුගණක I

69%

of Grade 10 term tests include it (37 of 54)

174

marks in past papers (50 questions)

11 min

full lesson with quick checks

🎯 What you'll learn

Name the base, index and power in aⁿ = b
Change index form to logarithm form and back
Find simple logarithms, including logₐ 1, logₐ a and lg
Use the product and quotient laws of logarithms
Solve simple logarithm equations

Doubling bacteria

Milk left out on a hot Kurunegala afternoon: the bacteria in it double every 20 minutes. How many doublings until there are 1024 times as many? You need the index that turns 2 into 1024. That index has a name: a logarithm.

Base, index, power

In 2¹⁰ = 1024:• 2 is the base• 10 is the index• 1024 is the power (the value)A logarithm asks: 'what index do I put on the base to get this number?'So log₂ 1024 = 10, read 'log of 1024 to base 2 is 10'.

🧮 Index form ⟷ logarithm form

The base stays the base. The index becomes the answer of the log. Use it every time you switch between the two forms.

aⁿ = b ⟺ logₐ b = n (a > 0, a ≠ 1)

Follow the colours

Base 3 (blue) goes to the bottom of 'log'. 81 (red) sits after it. The index 4 (amber) becomes the answer.

✏️ Switching forms

(i) Write 10³ = 1000 and 5⁻² = 1/25 in logarithm form. (ii) Write log₂ 64 = 6 in index form.

110³ = 1000 → log₁₀ 1000 = 3, usually written lg 1000 = 3
25⁻² = 1/25 → log₅ (1/25) = −2. A log can be negative.
3log₂ 64 = 6 → base 2, index 6, so 2⁶ = 64

✅ (i) lg 1000 = 3, log₅ (1/25) = −2 (ii) 2⁶ = 64

Finding a logarithm

Write the number as a power of the base.log₃ 81: 81 = 3⁴, so log₃ 81 = 4log₂ (1/8): 1/8 = 2⁻³, so the answer is −3log₂ √8: √8 = 2^(3/2), so the answer is 3/2Know these by heart: 2, 4, 8, 16, 32, 64; 3, 9, 27, 81, 243; 5, 25, 125, 625.

⚠️ See 81 as 3⁴

In the 2017 and 2019 O/L papers many candidates could not write 81 as a power of 3, so they could not find log₃ 81. Others wrote 9 = 3 × 3 but never 3². Break the number down by the base until you reach 1: 81 → 27 → 9 → 3 → 1. Four divisions, so 81 = 3⁴.

Three facts to remember

• logₐ a = 1, because a¹ = a. So log₇ 7 = 1.• logₐ 1 = 0, because a⁰ = 1. So log₅ 1 = 0.• lg means log to base 10: lg 10 = 1, lg 100 = 2, lg 0.01 = −2.The fact lg 10 = 1 turns up in almost every log question.

Why logs add

Multiplying powers of the same base adds the indices: 2³ × 2⁴ = 2⁷.A log is an index, so logs follow the same rule:log₂ (8 × 16) = log₂ 8 + log₂ 16 = 3 + 4 = 7Dividing subtracts the indices, so:log₂ (32 ÷ 4) = 5 − 2 = 3

🧮 Laws of logarithms

The log of a product is the sum of the logs; the log of a quotient is the difference. All the logs must have the same base.

logₐ (xy) = logₐ x + logₐ y logₐ (x/y) = logₐ x − logₐ y

✏️ Simplify without tables

Find the value of lg 25 + lg 8 − lg 2.

1+ means multiply, − means divide: lg (25 × 8 ÷ 2)
225 × 8 ÷ 2 = 100, so it is lg 100
3100 = 10², so lg 100 = 2

✅ 2

⚠️ Laws that don't exist

lg (a + b) is not lg a + lg b. The law is for a product: lg (ab) = lg a + lg b.lg 8 ÷ lg 2 is not lg 4. Only lg 8 − lg 2 = lg 4.In one O/L paper only 4% could simplify a log expression using lg 10 = 1. Check each step against the two laws.

📌 Remember

• aⁿ = b ⟺ logₐ b = n: the base stays, the index is the log.• To find a log, write the number as a power of the base.• logₐ a = 1, logₐ 1 = 0, lg means base 10, lg 10 = 1.• log (xy) = log x + log y; log (x/y) = log x − log y.• To solve, combine into one log, then change to index form.

🧾 Quick revision sheet

aⁿ = b in logarithm form?	logₐ b = n
What does lg mean?	Logarithm to base 10. lg 1000 = 3.
logₐ 1 = ?	0, because a⁰ = 1.
logₐ a = ?	1, because a¹ = a.
Product law	logₐ (xy) = logₐ x + logₐ y
Quotient law	logₐ (x/y) = logₐ x − logₐ y
log₃ 81 = ?	4, because 3⁴ = 81.
lg 5 in terms of lg 2?	lg 5 = lg 10 − lg 2 = 1 − lg 2.