Exponentials and Logarithms | 指数与对数

📚 Exponentials and Logarithms | 指数与对数

Exponential functions and logarithms are among the most powerful tools in A-Level mathematics. They appear in finance, biology, physics and statistics, and they are essential for solving equations where the unknown appears in a power.

指数函数与对数是 A-Level 数学中最强大的工具之一。它们出现在金融、生物、物理和统计等领域，也是求解未知量位于指数位置上的方程的关键。

1. Why Exponentials and Logarithms Matter | 为什么指数与对数很重要

An exponential function is written in the form f(x) = aˣ, where a is a positive constant not equal to 1. The variable x is the exponent, which makes exponential growth fundamentally different from polynomial growth.

指数函数写成 f(x) = aˣ 的形式，其中 a 是不等于 1 的正数常数。自变量 x 位于指数位置，这使得指数增长本质上不同于多项式增长。

Logarithms are the inverse operation of exponentiation. If aˣ = b, then x = logₐ(b). Understanding both concepts together allows you to solve exponential equations and model real-world situations.

对数是指数运算的逆运算。如果 aˣ = b，那么 x = logₐ(b)。将这两个概念结合起来，你就能求解指数方程，并对现实世界中的情境建立模型。

In Edexcel A-Level Maths, this topic is vital for Pure Mathematics, and it also supports work in Statistics and Mechanics.

在 Edexcel A-Level 数学中，这个专题对纯数学至关重要，同时也为统计与力学部分的学习提供支持。

2. Exponential Functions and Their Graphs | 指数函数及其图像

The graph of y = aˣ always passes through (0, 1), because a⁰ = 1 for any positive a. When a > 1, the graph rises from left to right and is called exponential growth.

y = aˣ 的图像始终经过 (0, 1)，因为任何正数 a 都有 a⁰ = 1。当 a > 1 时，图像从左向右上升，称为指数增长。

When 0 < a < 1, the graph falls from left to right, and this is called exponential decay. In both cases, the x-axis is a horizontal asymptote.

当 0 < a < 1 时，图像从左向右下降，这称为指数衰减。在这两种情况下，x 轴都是水平渐近线。

For example, y = 2ˣ doubles as x increases by 1. Its table of values shows rapid growth:

例如，y = 2ˣ 在 x 每增加 1 时变为原来的 2 倍。其函数值表显示快速增长：

x	-2	-1	0	1	2	3
y = 2ˣ	1/4	1/2	1	2	4	8

The key property of exponential functions is that the rate of change is proportional to the current value.

指数函数的关键性质是：变化率与当前值成正比。

3. The Definition of the Logarithm | 对数的定义

The logarithm is defined as the inverse of an exponential. For a > 0, a ≠ 1 and x > 0, the statement logₐ(x) = y is equivalent to aʸ = x.

对数被定义为指数的逆运算。对于 a > 0，a ≠ 1 且 x > 0，等式 logₐ(x) = y 等价于 aʸ = x。

logₐ(x) = y ⇔ aʸ = x

This definition is often called the “logarithmic identity”. The number a is called the base, and x is called the argument.

这个定义通常称为“对数恒等式”。数 a 称为底数，x 称为真数。

For example, log₂(8) = 3 because 2³ = 8. Similarly, log₁₀(1000) = 3 because 10³ = 1000.

例如，log₂(8) = 3，因为 2³ = 8。类似地，log₁₀(1000) = 3，因为 10³ = 1000。

Two special results follow directly from the definition: logₐ(1) = 0 and logₐ(a) = 1.

由定义可以直接得到两个特殊结果：logₐ(1) = 0 和 logₐ(a) = 1。

4. Laws of Logarithms | 对数的运算法则

Logarithms satisfy three important algebraic laws. These laws allow you to simplify expressions and solve equations.

对数满足三条重要的代数运算法则。这些法则帮助你化简表达式并求解方程。

Product law: logₐ(mn) = logₐ(m) + logₐ(n)
Quotient law: logₐ(m/n) = logₐ(m) – logₐ(n)
Power law: logₐ(mⁿ) = n logₐ(m)

积法则：logₐ(mn) = logₐ(m) + logₐ(n)

商法则：logₐ(m/n) = logₐ(m) – logₐ(n)

幂法则：logₐ(mⁿ) = n logₐ(m)

These laws are true for all positive values of m and n, and for any positive base a ≠ 1.

这些法则对所有正数 m、n 以及任意正底数 a ≠ 1 都成立。

For example, log₃(6) + log₃(5) = log₃(30), and 2 log₅(4) = log₅(4²) = log₅(16).

例如，log₃(6) + log₃(5) = log₃(30)，且 2 log₅(4) = log₅(4²) = log₅(16)。

logₐ(mⁿ) = n logₐ(m)

A common exam mistake is to apply these laws to addition or subtraction inside the argument, such as writing logₐ(m + n) = logₐ(m) + logₐ(n). This is not true.

一个常见考试错误是将法则错误地用于真数内部的加法或减法，例如写成 logₐ(m + n) = logₐ(m) + logₐ(n)。这是错误的。

5. Changing the Base | 换底公式

Sometimes you need to evaluate a logarithm with a base that is not available on your calculator. The change-of-base formula solves this problem.

有时你需要计算计算器上不存在的底数的对数。换底公式可以解决这个问题。

logₐ(b) = log꜀(b) / log꜀(a)

In practice, most calculators only have keys for log₁₀ and logₑ, so you can write:

实际上，大多数计算器只有 log₁₀ 和 logₑ 键，因此你可以写成：

logₐ(b) = log₁₀(b) / log₁₀(a) = ln(b) / ln(a)

For example, log₂(10) = ln(10) / ln(2) ≈ 3.3219.

例如，log₂(10) = ln(10) / ln(2) ≈ 3.3219。

The change-of-base formula is useful when solving equations and when rewriting logarithms into a common base for comparison.

换底公式在解方程以及将不同底数的对数改写为共同底数以进行比较时非常有用。

6. Solving Exponential Equations | 解指数方程

To solve an equation where the unknown appears in an exponent, take logarithms of both sides. This turns the exponential equation into a linear equation.

要解未知量出现在指数中的方程，可以对等式两边取对数。这样就把指数方程转化为线性方程。

Example: Solve 3ˣ = 20.

例：解 3ˣ = 20。

ln(3ˣ) = ln(20)

x ln 3 = ln 20

x = ln 20 / ln 3 ≈ 2.727

You may use ln or log₁₀; the answer is the same. If the equation has more than one exponential term, try collecting terms on one side first.

你可以使用 ln 或 log₁₀，答案相同。如果方程中有多个指数项，请先尝试将同类项合并到一边。

For equations of the form a f(x) = b g(x), take logs of both sides and use the power law to bring the exponents down.

对于形如 a f(x) = b g(x) 的方程，对两边取对数，并使用幂法则将指数提下来。

2ⁿ ⁺ ¹ = 3²ⁿ ⁻ ² → (n + 1) ln 2 = (2n – 2) ln 3

Then expand the brackets and solve linear equation in n.

然后展开括号，解关于 n 的线性方程。

7. Natural Logarithms and e | 自然对数和 e

The number e is an irrational constant approximately equal to 2.71828. It is the most important base for exponential functions in A-Level Maths.

自然常数 e 是一个无理数，约等于 2.71828。在 A-Level 数学中，它是最重要的指数函数底数。

The natural logarithm, written ln x, is the logarithm with base e. Therefore y = ln x means x = eʸ.

自然对数记作 ln x，是以 e 为底的对数。因此 y = ln x 表示 x = eʸ。

ln(e) = 1, ln(1) = 0, e^{ln x} = x, ln(eˣ) = x

The function y = eˣ is its own derivative. This special property makes e extremely useful in calculus, particularly in solving differential equations.

函数 y = eˣ 的导数等于其自身。这一特殊性质使 e 在微积分中极其有用，尤其在解微分方程时。

In Edexcel exams, you should be comfortable with graphs of y = eˣ and y = ln x. They are reflections of each other in the line y = x.

在 Edexcel 考试中，你必须熟悉 y = eˣ 和 y = ln x 的图像。它们关于直线 y = x 互为镜像。

8. Exponential Modelling | 指数模型

Many real-world quantities change at a rate proportional to their current value. These quantities can be modelled using exponential functions of the form N(t) = N₀ eᵏᵗ.

许多现实世界中的量以与当前值成正比的变化率变化。这些量可以用形如 N(t) = N₀ eᵏᵗ 的指数函数建模。

If k > 0, the model describes exponential growth, for example population growth or compound interest. If k < 0, it describes exponential decay, for example radioactive decay or cooling.

如果 k > 0，该模型描述指数增长，例如人口增长或复利增长。如果 k < 0，则描述指数衰减，例如放射性衰变或冷却过程。

The value of N₀ is the initial quantity at t = 0, because N(0) = N₀ e⁰ = N₀.

N₀ 是 t = 0 时的初始量，因为 N(0) = N₀ e⁰ = N₀。

To find the time at which a quantity reaches a certain value, substitute the known values and take ln of both sides.

要求某个量达到特定值所需的时间，代入已知值并对两边取 ln。

For example, if a population grows according to P(t) = 100 e^{0.02t}, the time to double satisfies 200 = 100 e^{0.02t}, so e^{0.02t} = 2, hence 0.02t = ln 2 and t = 50 ln 2 ≈ 34.7 years.

例如，若人口按 P(t) = 100 e^{0.02t} 增长，则翻倍时间满足 200 = 100 e^{0.02t}，即 e^{0.02t} = 2，因此 0.02t = ln 2，得 t = 50 ln 2 ≈ 34.7 年。

9. Logarithmic Graphs and Linearisation | 对数图像与线性化

Data that follows an exponential law can be transformed into a straight-line graph by plotting ln y against x. This process is called linearisation.

如果数据服从指数规律，可以通过绘制 ln y 关于 x 的图像将其转化为直线图。这一过程称为线性化。

If y = a bˣ, then taking ln of both sides gives ln y = ln a + x ln b. This has the form Y = mX + c, where Y = ln y, X = x, m = ln b and c = ln a.

如果 y = a bˣ，那么对两边取 ln 得到 ln y = ln a + x ln b。这具有 Y = mX + c 的形式，其中 Y = ln y，X = x，m = ln b，c = ln a。

ln y = ln a + x ln b

Similarly, if y = a xⁿ, then plotting ln y against ln x gives a straight line with gradient n and intercept ln a.

类似地，如果 y = a xⁿ，那么绘制 ln y 关于 ln x 的图像会得到一条斜率为 n、截距为 ln a 的直线。

This technique is often used in exam questions to determine unknown constants from given data or a straight-line graph.

这种技巧常用于考试题中，通过给定数据或直线图像确定未知常数。

10. Differentiation of eˣ and ln x | eˣ 和 ln x 的求导

Two derivatives must be memorised for Edexcel Pure Maths:

在 Edexcel 纯数学中，有两个导数必须牢记：

d/dx(eˣ) = eˣ, d/dx(ln x) = 1/x

Using the chain rule, you can differentiate composite functions. If y = e^{kx}, then dy/dx = k e^{kx}. If y = ln(kx), then dy/dx = 1/x.

利用链式法则，你可以对复合函数求导。若 y = e^{kx}，则 dy/dx = k e^{kx}。若 y = ln(kx)，则 dy/dx = 1/x。

For example, if y = e^{2x}, then dy/dx = 2e^{2x}. If y = ln(3x), then dy/dx = 3/(3x) = 1/x.

例如，若 y = e^{2x}，则 dy/dx = 2e^{2x}。若 y = ln(3x)，则 dy/dx = 3/(3x) = 1/x。

Using the product and quotient rules, you can differentiate expressions such as x² eˣ or eˣ/x.

利用乘积法则和商法则，你可以对 x² eˣ 或 eˣ/x 这样的表达式进行求导。

11. Exam-Style Worked Example | 考试型例题

Solve the equation 5^{x+1} = 2^{3x}, giving your answer to 3 significant figures.

解方程 5^{x+1} = 2^{3x}，答案保留三位有效数字。

Take ln of both sides:

对方程两边取 ln：

ln(5^{x+1}) = ln(2^{3x})

(x + 1) ln 5 = 3x ln 2

Expand the left side:

展开左边：

x ln 5 + ln 5 = 3x ln 2

Collect terms with x:

将所有含 x 的项移到一边：

ln 5 = 3x ln 2 – x ln 5

ln 5 = x(3 ln 2 – ln 5)

Therefore:

因此：

x = ln 5 / (3 ln 2 – ln 5) ≈ 1.60

Always substitute your answer back to check that both sides are approximately equal.

一定要将答案代回原方程检查两边是否近似相等。

12. Summary and Key Takeaways | 总结与要点

The exponential function and the logarithmic function are inverses of each other. The core definition is logₐ(x) = y ⇔ aʸ = x.

指数函数和对数函数互为反函数。核心定义是 logₐ(x) = y ⇔ aʸ = x。

An exponential graph has the form y = aˣ and passes through (0, 1).
The x-axis is a horizontal asymptote.
The three laws of logarithms are product, quotient and power.
Use the change-of-base formula when the base is not available on a calculator.
Take logarithms of both sides to solve exponential equations.
e is the natural base, and ln is the natural logarithm.
d/dx(eˣ) = eˣ and d/dx(ln x) = 1/x.

指数图像的形式为 y = aˣ 且经过 (0, 1)。

x 轴是水平渐近线。

对数的三大运算法则是积法则、商法则和幂法则。

当底数在计算器上不可用时，使用换底公式。

通过对方程两边取对数来解指数方程。

e 是自然底数，ln 是自然对数。

d/dx(eˣ) = eˣ，且 d/dx(ln x) = 1/x。

Master these ideas, and you will solve exponential and logarithmic problems quickly and accurately in the Edexcel exam.

掌握这些要点，你就能在 Edexcel 考试中快速准确地解决指数与对数问题。

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Exponentials and Logarithms | 指数与对数