Why bother with indices and logarithms?
Ever wondered how scientists calculate huge numbers without writing endless zeros? Or how your phone figures out the strength of a signal? Both tricks use the magic of indices (exponents) and logarithms. Let’s crack them together.
💡 In Simple Words: An index tells you how many times to multiply a number by itself. A logarithm tells you the opposite – how many times you need to multiply the base to get a certain number. Think of it like a recipe: the index is the number of scoops, the log is the number of scoops you need to reach a taste.
What is an Index (Exponent)?
An index (or exponent) is the small raised number you see after a base, like the 3 in 2³. It means “multiply the base (2) by itself three times”: 2 × 2 × 2 = 8. If the base is a variable, say a, then a⁴ means a × a × a × a.
Key Laws of Indices
These are shortcuts that let you simplify messy exponent problems. Below each law, you’ll find a short example.
- Product Law: aⁿ × aᵐ = aⁿ⁺ᵐ. When the bases are the same, just add the exponents.
Example: 3² × 3³ = 3⁵ = 243. - Quotient Law: aⁿ ÷ aᵐ = aⁿ⁻ᵐ. Same base, subtract the exponents.
Example: 5⁶ ÷ 5² = 5⁴ = 625. - Power of a Power Law: (aⁿ)ᵐ = aⁿᵐ. Multiply the exponents.
Example: (2³)² = 2⁶ = 64. - Power of a Product Law: (ab)ⁿ = aⁿ bⁿ. Raise each factor to the exponent separately.
Example: (4×5)² = 4² × 5² = 16 × 25 = 400. - Power of a Quotient Law: (a/b)ⁿ = aⁿ / bⁿ. Raise numerator and denominator individually.
Example: (6/3)³ = 6³ / 3³ = 216 / 27 = 8. - Zero Exponent Law: a⁰ = 1 (provided a ≠ 0). Anything raised to zero becomes 1.
Example: 9⁰ = 1. - Negative Exponent Law: a⁻ⁿ = 1 / aⁿ. Flip the fraction and make the exponent positive.
Example: 2⁻³ = 1 / 2³ = 1/8.
What is a Logarithm?
A logarithm answers the question: "To what power must we raise a certain base to get a given number?" It’s written as log_b a, read as "log base b of a". If bˣ = a, then log_b a = x. For example, because 10³ = 1000, we write log₁₀ 1000 = 3. Think of it like a ladder: the base is the step size, the number is how high you want to climb, and the log tells you how many steps you need.
Important Logarithm Rules
Just like indices, logarithms have their own set of handy shortcuts.
- Product Rule: log_b (mn) = log_b m + log_b n. The log of a product equals the sum of the logs.
Example: log₂ (8×4) = log₂ 8 + log₂ 4 = 3 + 2 = 5. - Quotient Rule: log_b (m/n) = log_b m – log_b n. The log of a division equals the difference of the logs.
Example: log₅ (125/5) = log₅ 125 – log₅ 5 = 3 – 1 = 2. - Power Rule: log_b (mⁿ) = n·log_b m. Bring the exponent down as a multiplier.
Example: log₃ (27²) = 2·log₃ 27 = 2·3 = 6. - Change of Base Formula: log_b a = log_k a / log_k b. You can switch to any new base k (often 10 or e) to simplify calculations.
Example: log₄ 16 = log₁₀ 16 / log₁₀ 4 ≈ 1.2041 / 0.6021 = 2. - Log of 1: log_b 1 = 0 because any base raised to 0 equals 1.
Example: log₇ 1 = 0. - Log of Base: log_b b = 1 because the base to the power 1 gives itself.
Example: log₈ 8 = 1.
Quick Comparison: Index Laws vs Log Rules
| Concept | Index Law | Logarithm Rule |
|---|---|---|
| Multiplying same base | aⁿ × aᵐ = aⁿ⁺ᵐ | log_b (mn) = log_b m + log_b n |
| Dividing same base | aⁿ ÷ aᵐ = aⁿ⁻ᵐ | log_b (m/n) = log_b m – log_b n |
| Power of a power | (aⁿ)ᵐ = aⁿᵐ | log_b (mⁿ) = n·log_b m |
| Zero exponent | a⁰ = 1 | log_b 1 = 0 |
| Negative exponent | a⁻ⁿ = 1 / aⁿ | log_b (1/aⁿ) = –n·log_b a |
How to Use These Laws in Exams
When a question looks messy, ask yourself: "Can I rewrite it using a law so the numbers get smaller?" Often, the trick is to spot a common base or a product that matches a rule.
- Identify if the expression has the same base – then apply product, quotient, or power‑of‑a‑power laws.
- If a log appears, check whether the argument (the number inside the log) can be broken into factors or powers.
- Never forget the zero and negative rules – they turn seemingly impossible numbers into simple fractions.
📝 Likely Exam Questions
- Simplify: (2⁴ × 2³) ÷ 2².
Answer: Use product law → 2⁴⁺³ = 2⁷. Then quotient law → 2⁷⁻² = 2⁵ = 32. - Write as a single power: (5³)² × 5⁻⁴.
Answer: Power‑of‑a‑power → 5⁶. Then product law → 5⁶⁻⁴ = 5² = 25. - Evaluate: log₃ 81.
Answer: 81 = 3⁴, so log₃ 81 = 4 (using definition of log). - Simplify using log rules: log₂ 8 + log₂ 4 – log₂ 2.
Answer: Convert each: log₂ 8 = 3, log₂ 4 = 2, log₂ 2 = 1. So 3 + 2 – 1 = 4. - Apply change of base: Express log₅ 125 as a common log (base 10).
Answer: log₅ 125 = log₁₀ 125 / log₁₀ 5 ≈ 2.0969 / 0.6990 ≈ 3.