📚 Exponential Functions and Logarithms | 指数函数与对数
Exponential functions and logarithms are central tools in A-Level Mathematics. They model rapid growth, decay, and many real-world processes, and they appear throughout the Edexcel specification. This revision guide summarises the key definitions, laws, graphs, differentiation, integration and modelling techniques you need to master.
指数函数与对数是 A-Level 数学的核心工具。它们可以描述快速增长、衰减以及许多现实过程，并贯穿 Edexcel 考试大纲。本复习指南总结你需要掌握的关键定义、法则、图像、微分、积分和建模技巧。
1. Exponential Functions: The Basics | 指数函数基础
An exponential function has the form y = ax, where a is a positive constant and a ≠ 1. The variable x appears in the exponent, which makes the rate of change proportional to the current value.
指数函数形如 y = ax，其中 a 是正常数且 a ≠ 1。变量 x 出现在指数位置，因此变化率与当前值成正比。
For any base a > 0, the graph passes through (0, 1), lies entirely above the x-axis, and has a horizontal asymptote y = 0. If a > 1, the function is increasing; if 0 < a < 1, it is decreasing.
对于任意底数 a > 0，图像都经过 (0, 1)，完全位于 x 轴上方，并以 y = 0 为水平渐近线。若 a > 1，函数递增；若 0 < a < 1，函数递减。
- Domain: all real numbers, x ∈ ℝ
- Range: y > 0
- One-to-one function, so it has an inverse
定义域为全体实数 x ∈ ℝ，值域为 y > 0。它是一一对应函数，因此存在反函数。
2. Introducing Logarithms | 对数的引入
The logarithm is the inverse of an exponential function. The statement y = ax is equivalent to x = logₐ y, read as ‘log base a of y’. It answers the question: to what power must a be raised to obtain y?
对数是指数函数的反函数。等式 y = ax 等价于 x = logₐ y，读作“以 a 为底 y 的对数”。它回答的问题是：a 要升到多少次幂才能得到 y？
For example, log₂ 8 = 3 because 2³ = 8, and log₁₀ 0.001 = -3 because 10⁻³ = 0.001.
例如，log₂ 8 = 3，因为 2³ = 8；log₁₀ 0.001 = -3，因为 10⁻³ = 0.001。
Key restrictions: logₐ x is defined only for x > 0, and the base a must satisfy a > 0 and a ≠ 1.
关键限制：logₐ x 只有在 x > 0 时有定义，底数 a 必须满足 a > 0 且 a ≠ 1。
3. The Laws of Logarithms | 对数运算法则
The laws of logarithms allow you to combine, expand and simplify logarithmic expressions. These are derived directly from index laws.
对数运算法则用于合并、展开和化简对数表达式。这些法则直接由指数法则推导而来。
| Law | Chinese meaning |
|---|---|
| logₐ x + logₐ y = logₐ(xy) | 乘法法则：对数相加等于真数相乘 |
| logₐ x – logₐ y = logₐ(x/y) | 除法法则：对数相减等于真数相除 |
| n logₐ x = logₐ(xⁿ) | 幂法则：系数移到真数的指数位置 |
| logₐ 1 = 0, logₐ a = 1 | 特殊值：1 的对数为 0，底数的对数为 1 |
| a^(logₐ x) = x | 指数与对数互逆还原 |
These laws are useful for solving equations and for changing the form of an expression to make differentiation or integration easier.
这些法则在解方程以及改变表达式形式以简化微分或积分时非常有用。
4. Solving Exponential Equations | 解指数方程
To solve equations where the unknown is in the exponent, take logarithms of both sides. You may use any base, but log₁₀ and ln are common because calculators provide them directly.
当未知数出现在指数中时，可对方程两边取对数。可以使用任意底数，但常用 log₁₀ 和 ln，因为计算器可以直接求出。
Example: Solve 3x = 20.
示例：求解 3x = 20。
3x = 20
log 3x = log 20
x log 3 = log 20
x = log 20 ÷ log 3 ≈ 2.727
If the equation contains e, use the natural logarithm ln as the inverse of ex. For example, e2x+1 = 5 gives 2x + 1 = ln 5.
如果方程含有 e，可用自然对数 ln 作为 ex 的逆运算。例如，e2x+1 = 5 可得 2x + 1 = ln 5。
5. Solving Logarithmic Equations | 解对数方程
For logarithmic equations, use the laws of logarithms to combine terms, then rewrite in exponential form. Always check that the original arguments are positive; some algebraic solutions may be invalid.
解对数方程时，先用对数法则合并项，再改写成指数形式。务必检查原式的真数是否为正；有些代数解可能无效。
Example:Solve log₂(x) + log₂(x – 2) = 3.
示例：求解 log₂(x) + log₂(x – 2) = 3。
log₂[x(x – 2)] = 3
x(x – 2) = 2³ = 8
x² – 2x – 8 = 0
(x – 4)(x + 2) = 0
x = 4 or x = -2
x = -2 must be rejected because log₂(-2) is undefined. The only valid solution is x = 4.
x = -2 必须舍去，因为 log₂(-2) 无定义。唯一有效解是 x = 4。
6. The Natural Exponential Function eˣ and Natural Logarithm ln x | 自然指数函数 eˣ 与自然对数 ln x
The natural exponential function is y = ex, where e ≈ 2.71828. It is the only
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