ICSE Class 9 Maths: Laws of Indices & Logarithms

Why the Laws of Indices and Logarithms Matter

Ever wonder how to shrink huge numbers like 2⁸ or solve equations with exponents quickly? That's where the laws of indices and logarithms step in. They turn messy math into neat tricks you can use in exams and real life.

💡 In Simple Words: Indices (or exponents) tell us how many times to multiply a number by itself. Logarithms answer the opposite question – they ask, “To what power must we raise a base to get this number?” Knowing the rules lets you jump between the two in a flash.

Key Definitions

Index (or exponent) – the small number written up‑right that says how many copies of the base we multiply. For example, in 3³, the ³ is the index.

Logarithm – the power you need to raise a base (commonly 10 or e) to reach a given number. Written as log₁₀ 100 = 2 because 10² = 100.

Laws of Indices (Exponent Rules)

These rules work like shortcuts when you’re dealing with the same base.

• Product law: aᵐ × aⁿ = aᵐ⁺ⁿ. Multiply same bases, add the indices. Think of stacking LEGO bricks – each brick adds height.
• Quotient law: aᵐ ÷ aⁿ = aᵐ⁻ⁿ. Divide same bases, subtract the indices. Like taking away floors from a building.
• Power of a power: (aᵐ)ⁿ = aᵐⁿ. Raise a power to another power, multiply the indices. Imagine a recipe that doubles a double‑batch.
• Power of a product: (ab)ⁿ = aⁿ bⁿ. Distribute the index to each factor. Like sharing candies equally among kids.
• Zero index: a⁰ = 1 (provided a ≠ 0). Anything to the power zero becomes one – like an empty basket still counts as one basket.
• Negative index: a⁻ⁿ = 1 / aⁿ. A negative exponent flips the fraction. Think of walking backwards – you end up on the opposite side.

Worked Example – Using Index Laws

Simplify 2⁴ × 2⁻² ÷ 2³.

Step 1: Combine the product and quotient using the product and quotient laws.

2⁴ × 2⁻² = 2⁴⁺⁻² = 2².

Now divide by 2³: 2² ÷ 2³ = 2²⁻³ = 2⁻¹.

Finally, apply the negative index rule: 2⁻¹ = 1 / 2.

Answer: ½.

Logarithm Rules (Properties of Logarithms)

Log rules are the mirror image of index rules because a logarithm is just the inverse of an exponent.

• Product rule: logₐ (xy) = logₐ x + logₐ y. Log of a product becomes sum of logs. Like turning a combined score into individual scores.
• Quotient rule: logₐ (x / y) = logₐ x – logₐ y. Log of a division becomes difference of logs.
• Power rule: logₐ (xⁿ) = n·logₐ x. Bring the exponent down as a multiplier. Imagine unwrapping a gift; the size of the box tells you the number of layers.
• Change‑of‑base rule: log_b x = (log_a x) / (log_a b). Switch the base to something more convenient, usually 10 or e.

Worked Example – Using Log Rules

Simplify log₂ (8) + log₂ (4).

Both numbers are powers of 2: 8 = 2³, 4 = 2².

log₂ 8 = 3 and log₂ 4 = 2, so the sum is 5.

Alternatively, use the product rule: log₂ (8·4) = log₂ 32 = 5 because 2⁵ = 32.

Comparison Table: Index Laws vs Logarithm Rules

Tips for Quick Exam Solving

• Always write down the base when you use a rule – it prevents mixing up different bases.
• Convert numbers to the same base first if the rule involves the same base (e.g., change 27 to 3³).
• For logarithms, if the base isn’t given, assume base 10 (common log) or e (natural log) as the question states.
• Check your answer by raising the base to the obtained exponent – a quick sanity test.

📝 Likely Exam Questions

1. Simplify (5³ × 5⁻¹) ÷ 5².
   Answer: 5³⁺⁻¹⁻² = 5⁰ = 1.
2. Evaluate log₃ 27 – log₃ 9.
   Answer: log₃ 27 = 3, log₃ 9 = 2, difference = 1.
3. Express log₁₀ 50 using the change‑of‑base rule with natural logs (ln).
   Answer: log₁₀ 50 = (ln 50) / (ln 10).
4. If 2ˣ = 16, find x using logarithms.
   Answer: Take log₂ both sides: x = log₂ 16 = 4.
5. Write the expression (3²·3⁴)⁻¹ as a single power of 3.
   Answer: Inside brackets: 3²⁺⁴ = 3⁶. Then apply negative exponent: (3⁶)⁻¹ = 3⁻⁶ = 1/3⁶.