Indices and Logarithms for CIE IGCSE | IGCSE 数学指数与对数考点精讲

📚 Indices and Logarithms for CIE IGCSE | IGCSE 数学指数与对数考点精讲

Indices and logarithms are fundamental concepts in the CIE IGCSE Mathematics syllabus (Extended). Understanding their rules and applications is crucial for simplifying expressions and solving complex equations. This guide covers all key points you need to master for the exam.

指数与对数是 CIE IGCSE 数学（拓展）课程中的基础概念。理解其运算法则和应用对于简化表达式及求解复杂方程至关重要。本文涵盖了你备考需要掌握的全部考点。

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1. Laws of Indices | 指数运算法则

Product Rule: aᵐ × aⁿ = aᵐ⁺ⁿ. When multiplying powers with the same base, add the exponents.

乘积法则： aᵐ × aⁿ = aᵐ⁺ⁿ。同底数幂相乘，指数相加。

Example: 2³ × 2⁴ = 2⁷ = 128.

例如：2³ × 2⁴ = 2⁷ = 128。

Quotient Rule: aᵐ ÷ aⁿ = aᵐ⁻ⁿ. When dividing powers with the same base, subtract the exponents.

商法则： aᵐ ÷ aⁿ = aᵐ⁻ⁿ。同底数幂相除，指数相减。

Example: 5⁶ ÷ 5² = 5⁴ = 625.

例如：5⁶ ÷ 5² = 5⁴ = 625。

Power Rule: (aᵐ)ⁿ = aᵐⁿ. To raise a power to another power, multiply the exponents.

幂的乘方法则： (aᵐ)ⁿ = aᵐⁿ。幂的乘方，指数相乘。

Example: (3²)⁴ = 3⁸ = 6561.

例如：(3²)⁴ = 3⁸ = 6561。

Power of a Product: (ab)ⁿ = aⁿbⁿ. A product raised to a power equals each factor raised to that power.

积的乘方： (ab)ⁿ = aⁿbⁿ。乘积的幂等于各因式幂的乘积。

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2. Zero and Negative Indices | 零指数与负指数

Zero Index: a⁰ = 1, provided a ≠ 0. Any non-zero number raised to the power of 0 equals 1.

零指数： a⁰ = 1 (a ≠ 0)。任何非零数的 0 次方都等于 1。

Example: 7⁰ = 1, (−2)⁰ = 1.

例如：7⁰ = 1，(−2)⁰ = 1。

Negative Index: a⁻ⁿ = 1 / aⁿ. A negative exponent indicates the reciprocal of the positive power.

负指数： a⁻ⁿ = 1 / aⁿ。负指数表示正指数幂的倒数。

Example: 2⁻³ = 1 / 2³ = 1/8.

例如：2⁻³ = 1 / 2³ = 1/8.

You can combine these rules: simplify (x⁻² y³)⁻¹ = x² y⁻³ = x² / y³.

可结合法则使用：化简 (x⁻² y³)⁻¹ = x² y⁻³ = x² / y³。

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3. Fractional Indices | 分数指数

Fractional index 1/n: a¹⁄ⁿ = ⁿ√a. The denominator n indicates the nth root.

分数指数 1/n： a¹⁄ⁿ = ⁿ√a。分母 n 表示 n 次方根。

Example: 9¹⁄² = √9 = 3; 64¹⁄³ = ³√64 = 4.

例如：9¹⁄² = √9 = 3；64¹⁄³ = ³√64 = 4。

Fractional index m/n: aᵐ⁄ⁿ = (ⁿ√a)ᵐ = ⁿ√(aᵐ). Raise to the power m and take the nth root (order does not matter).

分数指数 m/n： aᵐ⁄ⁿ = (ⁿ√a)ᵐ = ⁿ√(aᵐ)。先开 n 次方再 m 次方，或先乘方后开方。

Example: 8²⁄³ = (³√8)² = 2² = 4, also ³√(8²) = ³√64 = 4.

例如：8²⁄³ = (³√8)² = 2² = 4，也可 ³√(8²) = ³√64 = 4。

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4. Solving Simple Exponential Equations | 解简单指数方程

When both sides can be expressed as powers of the same base, equate the exponents.

两边能写成同底数幂时，直接令指数相等。

Example 1: 2ˣ = 16 → 2ˣ = 2⁴ → x = 4.

例 1：2ˣ = 16 → 2ˣ = 2⁴ → x = 4。

Example 2: 3²ˣ⁻¹ = 27 → 3²ˣ⁻¹ = 3³ → 2x − 1 = 3 → x = 2.

例 2：3²ˣ⁻¹ = 27 → 3²ˣ⁻¹ = 3³ → 2x − 1 = 3 → x = 2。

Example 3: 5ˣ⁺² = 1/25 → 5ˣ⁺² = 5⁻² → x + 2 = −2 → x = −4.

例 3：5ˣ⁺² = 1/25 → 5ˣ⁺² = 5⁻² → x + 2 = −2 → x = −4。

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5. Definition of Logarithms | 对数的定义

If aˣ = b, then x = logₐ b, where a > 0, a ≠ 1, and b > 0. The logarithm is the exponent to which the base must be raised to obtain the number.

若 aˣ = b，则 x = logₐ b，其中 a > 0, a ≠ 1, b > 0。对数是指数的逆运算。

Example: 2³ = 8 ←→ log₂ 8 = 3. “log base 2 of

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