📚 KS3 Maths: Indices and Logarithms Essentials | KS3 数学：指数与对数 考点精讲
Indices (or powers) are a shorthand way of expressing repeated multiplication, and logarithms help us solve equations involving indices. In this KS3 revision guide, we will explore the fundamental rules governing indices, learn how to work with negative and fractional indices, and discover the concept of logarithms as the inverse operation.
指数（或幂）是表示重复乘法的简化方式，而对数帮助我们解决涉及指数的方程。在这份KS3复习指南中，我们将探索支配指数的基本规则，学习如何处理负指数和分数指数，并了解作为逆运算的对数概念。
1. What are Indices? | 什么是指数？
by itself. In the expression 2³, 2 is the base and 3 is the index, meaning 2 × 2 × 2 = 8.
指数告诉我们一个底数自乘多少次。在表达式 2³ 中，2 是底数，3 是指数，表示 2 × 2 × 2 = 8。
The index 2 is called 'squared' and 3 is called 'cubed'. So 4² = 4 × 4 = 16, and 5³ = 5 × 5 × 5 = 125.
指数 2 称为“平方”，指数 3 称为“立方”。因此 4² = 4 × 4 = 16，5³ = 5 × 5 × 5 = 125。
Indices can be any integer, and later we will see that they can also be fractions or negative numbers.
指数可以是任何整数，后面我们会看到它们也可以是分数或负数。
2. Multiplication Rule | 乘法法则
When multiplying powers with the same base, we add the indices. The rule is: aᵐ × aⁿ = aᵐ⁺ⁿ (where a ≠ 0).
当同底数的幂相乘时，指数相加。法则为：aᵐ × aⁿ = aᵐ⁺ⁿ（a ≠ 0）。
ple, 2³ × 2⁴ = 2⁽³⁺⁴⁾ = 2⁷ = 128. We keep the base (2) and simply add the indices 3 and 4.
例如，2³ × 2⁴ = 2⁽³⁺⁴⁾ = 2⁷ = 128。我们保留底数（2）并将指数 3 和 4 相加。
rule only works when the bases are the same; you cannot combine 2³ × 3² using this method.
该法则仅当底数相同时适用；你不能用这种方法合并 2³ × 3²。
3. Division Rule | 除法法则
When dividing powers with the same base, we subtract the indices. The rule is: aᵐ ÷ aⁿ = aᵐ⁻ⁿ (a ≠ 0).
当同底数的幂相除时，指数相减。法则为：aᵐ ÷ aⁿ = aᵐ⁻ⁿ（a ≠ 0）。
÷ 5² = 5⁽⁵⁻²⁾ = 5³ = 125. Alternatively, we can think of it as cancelling common factors.
例如，5⁵ ÷ 5² = 5⁽⁵⁻²⁾ = 5³ = 125。我们也可以把它看作约去公因数。
Remember that the larger index always goes first, but the rule works even if m < n, giving a negative index (which we will cover soon).
记住通常用较大的指数减去较小的，但即使 m < n 该法则也有效，会得到负指数（稍后介绍）。
4. Power of a Power Rule | 幂的幂法则
When raising a power to another power, we multiply the indices. The rule is: (aᵐ)ⁿ = am×n.
当对一个幂再进行乘方时，指数相乘。法则为：(aᵐ)ⁿ = am×n。
Example: (3²)³ = 32×3 = 3⁶ = 729. First square 3 to get 9, then cube 9 gives 729, which matches 3⁶.
例子：(3²)³ = 32×3 = 3⁶ = 729。先将 3 平方得 9，再将 9 立方得 729，与 3⁶ 一致。
his rule is very useful when simplifying expressions with nested indices, like (x²)⁴ = x⁸.
这个法则在简化带有嵌套指数的表达式时非常有用，如 (x²)⁴ = x⁸。
5. Zero Index | 零指数
Any non-zero number raised to the power of zero equals 1. The rule is: a⁰ = 1 (a ≠ 0).
任何非零数的零次方都等于 1。规则是：a⁰ = 1（a ≠ 0）。
g the division rule: a³ ÷ a³ = a⁽³⁻³⁾ = a⁰ = 1, because any number divided by itself is 1.
这可以用除法法则来理解：a³ ÷ a³ = a⁽³⁻³⁾ = a⁰ = 1，因为任何数除以它自身都得 1。
Thus 10⁰ = 1, 5⁰ = 1, and even (‑3)⁰ = 1. Note that 0⁰ is undefined.
因此 10⁰ = 1，5⁰ = 1，甚至 (‑3)⁰ = 1。注意 0⁰ 是未定义的。
6. Negative Indices | 负指数
A negative index indicates the reciprocal of the positive power. The rule is: a⁻ⁿ = 1 / aⁿ (a ≠ 0).
负指数表示正指数幂的倒数。规则是：a⁻ⁿ = 1 / aⁿ（a ≠ 0）。
For example, 2⁻³ = 1 / 2³ = 1/8. Similarly, 10⁻² = 1/10² = 1/100 = 0.01.
例如，2⁻³ = 1 / 2³ = 1/8。类似地，10⁻² = 1/10² = 1/100 = 0.01。
en numerator and denominator by changing the sign of its index: a⁻ⁿ = 1/aⁿ and 1/a⁻ⁿ = aⁿ.
你也可以通过改变指数的符号在分子与分母之间移动因式：a⁻ⁿ = 1/aⁿ 以及 1/a⁻ⁿ = aⁿ。
7. Fractional Indices | 分数指数
A fractional index represents a root. The denominator of the fraction tells us the type of root: a1/n = ⁿ√a.
分数指数表示开方。分数的分母告诉我们开方次数：a1/n = ⁿ√a。
For instance, 9½ = √9 = 3, and 8⅓ = ∛8 = 2. The numerator then gives a power: am/n = ⁿ√aᵐ or (ⁿ√a)ᵐ.
例如，9½ = √9 = 3，8⅓ = ∛8 = 2。分子则给出乘方：am/n = ⁿ√aᵐ 或 (ⁿ√a)ᵐ。
So 16¾ = (⁴√16)³ = 2³ = 8. This idea combines roots and powers smoothly.
所以 16¾ = (⁴√16)³ = 2³ = 8。这一概念将开方和乘方巧妙地结合了起来。
8. Introduction to Logarithms | 对数入门
A logarithm is the inverse operation of raising to a power. If bˣ = y, then we write x = log_b y (read 'log base b of y').
对数是指数运算的逆运算。如果 bˣ = y，那么我们写作 x = log_b y（读作“以 b 为底 y 的对数”）。
garithm is base 10, written as log₁₀. For example, since 10² = 100, we have log₁₀ 100 = 2.
最常用的是以 10 为底的对数，写作 log₁₀。例如，因为 10² = 100，所以 log₁₀ 100 = 2。
Logarithms answer the question: 'To what power must we raise the base to obtain a given number?' So log₂ 8 = 3 because 2³ = 8.
对数回答的问题是：“底数需要几次方才能得到给定的数？”因此 log₂ 8 = 3，因为 2³ = 8。
9. Common Logarithms and Their Properties | 常用对数及其性质
Logarithms have useful properties that mirror the index laws. For base 10 logarithms, one key rule is: log₁₀ (AB) = log₁₀ A + log₁₀ B.
对数具有与指数定律相对应的有用性质。以 10 为底的对数，一个关键法则是：log₁₀ (AB) = log₁₀ A + log₁₀ B。
Let' : log₁₀ (100 × 10) = log₁₀ 1000 = 3, and log₁₀ 100 + log₁₀ 10 = 2 + 1 = 3. The rule holds.
我们来检验一下：log₁₀ (100 × 10) = log₁₀ 1000 = 3，而 log₁₀ 100 + log₁₀ 10 = 2 + 1 = 3。法则成立。
Similarly, the division rule for indices becomes log₁₀ (A / B) = log₁₀ A – log₁₀ B, and the power rule gives log₁₀ (Aⁿ) = n × log₁₀ A.
类似地，指数的除法法则变成 log₁₀ (A / B) = log₁₀ A – log₁₀ B，幂法则给出 log₁₀ (Aⁿ) =
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