1. Indices (Powers)

Indices (also called powers or exponents) show how many times a number (the base) is multiplied by itself.

Notation

If a is a number and n is a positive whole number, then aⁿ means a × a × ... × a (n times). Example: 3² = 3 × 3 = 9, and 5³ = 5 × 5 × 5 = 125.

Laws of Indices (simple rules)

• Product rule: a^m × aⁿ = a^m+n
• Quotient rule: a^m ÷ aⁿ = a^m−n (a ≠ 0)
• Power of a power: (a^m)ⁿ = a^m·n
• Zero exponent: a⁰ = 1 (for a ≠ 0)
• Negative exponent: a⁻ⁿ = 1 / aⁿ
• Fractional exponent: a^1/2 = √a, and a^m/n = (√[n]{a})^m

Quick tips

• When multiplying same base, add exponents. When dividing, subtract exponents.
• Negative exponent means reciprocal. Zero exponent gives 1.
• Fractional exponents connect with roots (e.g., a^1/3 is cube root of a).

2. Logarithms

Logarithms are the inverse of indices. They answer the question: "To what power must the base be raised to get a number?"

Definition

If b^y = x (with b>0, b≠1), then log_b(x) = y.

Example: Since 10² = 100, we write log₁₀(100) = 2. This is often written simply as log(100) = 2 (meaning base 10).

Laws of Logarithms

• Product rule: log_b(MN) = log_b(M) + log_b(N)
• Quotient rule: log_b(M / N) = log_b(M) − log_b(N)
• Power rule: log_b(M^k) = k · log_b(M)
• Change of base: log_a(x) = log_c(x) ÷ log_c(a) (useful when using a calculator)

Connection between indices and logs

Indices: b^y = x.   Logs: log_b(x) = y. They are the same information written differently.

Quick Visual Summary

Think: indices = "multiply many times", logs = "how many times?"

2⁴ = 16   ↔   log₂(16) = 4

10² = 100   ↔   log₁₀(100) = 2

Practice Questions

1. Simplify: 3² × 3³.
2. Write as a single power: (2³)².
3. Evaluate: 5⁰ and 10⁻¹.
4. Simplify using indices: 7⁵ ÷ 7².
5. Write √16 as an index.
6. Find log₁₀(1000).
7. Use log rules: If log(2)=0.3010 and log(5)=0.6990, find log(10).
8. Find log₂(32).

Try answering without a calculator for practice.

Answers

1. 3² × 3³ = 3⁵ = 243.
2. (2³)² = 2^3×2 = 2⁶ = 64.
3. 5⁰ = 1.   10⁻¹ = 1/10 = 0.1.
4. 7⁵ ÷ 7² = 7³ = 343.
5. √16 = 16^1/2 = 4.
6. log₁₀(1000) = 3 because 10³ = 1000.
7. log(10) = log(2×5) = log(2) + log(5) = 0.3010 + 0.6990 = 1. (So log₁₀(10) = 1.)
8. log₂(32) = 5 because 2⁵ = 32.

Final Tips

• Practice converting between roots and fractional indices: √a = a^1/2.
• Use index laws step by step — show exponents clearly when simplifying.
• Remember logs are just another way to record powers. If unsure, rewrite as b^y = x to find y.
• On calculators, log usually means log base 10. Use change-of-base for other bases.