📚 A-Level Edexcel Maths: Exponentials and Logarithms – Key Points | A-Level Edexcel 数学：指数与对数 考点精讲
Exponentials and logarithms are fundamental topics in the Edexcel A-Level Pure Mathematics syllabus. Understanding their definitions, properties, graphs, and applications in calculus and modelling is essential for success in exams. This revision guide covers key concepts, common question types, and essential techniques to help you master exponentials and logarithms.
指数与对数是 Edexcel A-Level 纯数学的核心内容。掌握它们的定义、性质、图像以及在微积分和建模中的应用是考试成功的关键。本文梳理了核心考点、常见题型与必备技巧，帮助你全面掌握指数与对数。
1. Exponential Functions | 指数函数
An exponential function is of the form y = aˣ, where a > 0 and a ≠ 1. The base a determines growth (a > 1) or decay (0 < a < 1). The most important exponential function in A-Level Maths is y = eˣ, where e ≈ 2.71828... is the natural base.
指数函数的形式为 y = aˣ，其中 a > 0 且 a ≠ 1。底数 a 决定函数的增长 (a > 1) 或衰减 (0 < a < 1)。A-Level 数学中最重要的是自然指数函数 y = eˣ，e ≈ 2.71828... 为自然常数。
Key properties: the domain is all real numbers (ℝ), the range is (0, ∞), the y-intercept is (0,1), and the x-axis is a horizontal asymptote. For y = eˣ, the gradient at any point equals the function value, leading to d/dx(eˣ) = eˣ.
主要性质：定义域为全体实数 (ℝ)，值域为 (0, ∞)，y 轴截距为 (0,1)，x 轴为水平渐近线。对于 y = eˣ，任一点的斜率等于该点的函数值，因此其导数 d/dx(eˣ) = eˣ。
2. The Natural Logarithm ln x | 自然对数
The natural logarithm, written as ln x, is the logarithm to base e. It is the inverse function of eˣ, meaning ln(eˣ) = x for all real x, and e^(ln x) = x for x > 0.
自然对数记作 ln x，是以 e 为底的对数。它是 eˣ 的反函数，即 ln(eˣ) = x 对一切实数 x 成立，且 e^(ln x) = x (x > 0)。
The graph of y = ln x passes through (1,0), has domain (0, ∞), range ℝ, and the y-axis is a vertical asymptote. Its derivative is d/dx (ln x) = 1/x.
y = ln x 的图像过点 (1,0)，定义域为 (0, ∞)，值域为 ℝ，y 轴为垂直渐近线。其导数为 d/dx (ln x) = 1/x。
3. Laws of Logarithms | 对数运算法则
For any base a > 0, a ≠ 1, and positive M, N, the three fundamental laws are:
对任意底数 a > 0, a ≠ 1，且正数 M, N，三条基本法则为：
logₐ(MN) = logₐ M + logₐ N
The product rule.
乘积法则。
logₐ(M/N) = logₐ M − logₐ N
The quotient rule.
商法则。
logₐ(Mⁿ) = n logₐ M
The power rule.
幂法则。
These laws apply to ln as well: ln(MN) = ln M + ln N, etc. Also useful: logₐ 1 = 0 and logₐ a = 1.
这些法则同样适用于自然对数：ln(MN) = ln M + ln N 等。此外，logₐ 1 = 0 且 logₐ a = 1。
4. Solving Exponential Equations | 解指数方程
Exponential equations often require taking logarithms of both sides. For an equation aˣ = b, take ln or log₁₀: x ln a = ln b, so x = ln b / ln a.
指数方程通常需要对两边取对数。对于 aˣ = b，可取自然对数：x ln a = ln b，因此 x = ln b / ln a。
If the equation involves e, use ln directly. Example: solve e²ˣ = 5. Take ln: 2x = ln 5, so x = (ln 5)/2.
若方程含有 e，可直接取 ln。例如：解 e²ˣ = 5。取对数：2x = ln 5，解得 x = (ln 5)/2。
For equations like 3²ˣ⁺¹ = 5ˣ, take ln: (2x+1) ln 3 = x ln 5, then solve for x.
对于 3²ˣ⁺¹ = 5ˣ 这类方程，取 ln 得 (2x+1) ln 3 = x ln 5，然后解出 x。
5. Solving Logarithmic Equations | 解对数方程
Logarithmic equations often require combining logs into a single logarithm and then exponentiating. Always check for extraneous solutions; arguments must be positive.
对数方程通常需要合并对数并取指数。务必检验增根，因为真数必须为正。
Example: solve ln(3x − 1) = 2. Exponentiate: 3x − 1 = e², so x = (e² + 1)/3. Check 3x − 1 > 0.
示例：解 ln(3x − 1) = 2。化为指数式：3x − 1 = e²，因此 x = (e² + 1)/3，并验证 3x − 1 > 0。
For logₐ(x + 2) + logₐ(x − 2) = logₐ 5, combine: logₐ((x+2)(x−2)) = logₐ 5, so x² − 4 = 5, x = ±3. x = −3 is extraneous because domain requires x > 2.
对于 logₐ(x + 2) + logₐ(x − 2) = logₐ 5，合并为 logₐ((x+2)(x−2)) = logₐ 5，得 x² − 4 = 5，x = ±3。由于真数要求 x > 2，x = −3 为增根。
6. Change of Base Formula | 换底公式
The change of base rule allows conversion between different bases: logₐ b = logₓ b / logₓ a, for any valid base c. Most commonly, we use base 10 or e.
换底公式可转换对数底数：logₐ b = logₓ b / logₓ a，其中 c 为任意有效底数。最常用的是自然对数或常用对数。
logₐ b = ln b / ln a = log₁₀ b / log₁₀ a
This is essential when evaluating logs with uncommon bases on a calculator.
当计算器无法直接计算非常用底数的对数时，此公式至关重要。
Example: log₂ 8 = ln 8 / ln 2 = 3. In exam questions, you may need to express an answer in terms of ln or log₁₀.
示例：log₂ 8 = ln 8 / ln 2 = 3。考试中可能需要用 ln 或 log₁₀ 表示答案。
7. Graphs of Exponential and Logarithmic Functions | 指数与对数函数图像
Know the shapes: y = aˣ (a>1) is an increasing curve through (0,1), asymptote at y = 0. y = aˣ (0
需熟记图像形状：y = aˣ (a>1) 为过 (0,1) 的递增曲线，渐近线为 y = 0。y = aˣ (0
The natural exponential y = eˣ and y = ln x are specific cases. Plotting points and showing transformations are common exam tasks.
自然指数 y = eˣ 与 y = ln x 是特例。作图与变换在考试中较为常见。
8. Transformations of Graphs | 图像变换
Apply the standard transformations to exponential and logarithmic graphs. For y = eˣ, y = eˣ⁺² shifts left by 2; y = 3eˣ stretches vertically by factor 3; y = e²ˣ compresses horizontally by 1/2.
对指数与对数图像应用常规变换。对于 y = eˣ，y = eˣ⁺² 向左平移 2 个单位；y = 3eˣ 垂直拉伸 3 倍；y = e²ˣ 水平压缩至 1/2。
For y = ln x, y = ln(x − 1) shifts right by 1; y = −ln x reflects in the x-axis; y = ln(2x) needs careful handling with property ln(2x) = ln 2 + ln x, so it is a vertical shift of ln x by ln 2.
对于 y = ln x，y = ln(x − 1) 向右平移 1；y = −ln x 关于 x 轴对称；y = ln(2x) 可写作 ln 2 + ln x，因此是 ln x 图像向上平移 ln 2。此类变形常作为考点。
9. Exponential Growth and Decay | 指数增长与衰减
Many real-world models use the form P = P₀ e^(kt), where k > 0 gives growth, k < 0 gives decay. P₀ is the initial value. The doubling time or half-life can be found by setting P = 2P₀ or ½P₀.
许多实际模型采用 P = P₀ e^(kt) 的形式，k > 0 表示增长，k < 0 表示衰减，P₀ 为初值。求翻倍时间或半衰期时，可令 P = 2P₀ 或 ½P₀。
Example: A population grows according to P = 100 e^(0.05t). Find time t when P = 200. Solve 200 = 100 e
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