Laws of Logarithms and Logarithmic Functions | 对数运算与对数函数

📚 Laws of Logarithms and Logarithmic Functions | 对数运算与对数函数

Logarithms are one of the most powerful tools in mathematics, transforming multiplication into addition, division into subtraction, and exponentiation into multiplication. In the IB Mathematics curriculum, a thorough understanding of logarithmic laws and logarithmic functions is essential for solving exponential equations, modelling real-world growth and decay, and tackling advanced calculus topics.

对数是数学中最强大的工具之一，它将乘法化为加法、除法化为减法、幂运算化为乘法。在IB数学课程中，深入理解对数运算法则和对数函数，是解指数方程、建立真实世界增长与衰减模型以及掌握高级微积分内容的关键基础。

1. Definition of Logarithms | 对数的定义

For any positive base a ≠ 1 and any positive number x, the logarithm of x to base a is defined as the exponent to which a must be raised to obtain x:

对于任意不等于 1 的正底数 a 以及任意正数 x，以 a 为底 x 的对数定义为：为使 a 的幂等于 x 所需的指数，即：

y = logₐ x ⇔ aʸ = x

This equivalence is the single most important idea to internalise: a logarithmic statement is merely an exponential statement written in disguise. For example, log₂ 8 = 3 because 2³ = 8.

这组等价关系是需要内化的最重要的核心概念：对数式本质上是指数式的另一种表达形式。例如，log₂ 8 = 3，因为 2³ = 8。

Two special bases deserve particular attention:

两个特殊底数值得特别关注：

Common logarithm: log₁₀ x, often written simply as log x.
Common logarithm | 常用对数：log₁₀ x，通常简写为 log x。
Natural logarithm: logₑ x, written as ln x, where e ≈ 2.71828.
自然对数：logₑ x，记作 ln x，其中 e ≈ 2.71828。

2. The Three Fundamental Laws | 三大基本法则

The power of logarithms lies in their ability to simplify algebraic operations. For positive numbers M, N and any real number r, the following laws hold:

对数的力量在于其化简代数运算的能力。对于正数 M、N 以及任意实数 r，下列法则成立：

Product Law 乘积法则

logₐ(M × N) = logₐ M + logₐ N

The logarithm of a product equals the sum of the logarithms. For instance, log₂(4 × 8) = log₂ 4 + log₂ 8 = 2 + 3 = 5, and indeed 2⁵ = 32 = 4 × 8.

乘积的对数等于各对数之和。例如，log₂(4 × 8) = log₂ 4 + log₂ 8 = 2 + 3 = 5，而确实 2⁵ = 32 = 4 × 8。

Quotient Law 商法则

logₐ(M ÷ N) = logₐ M − logₐ N

The logarithm of a quotient equals the difference of the logarithms. For example, log₁₀(1000 ÷ 10) = log₁₀ 1000 − log₁₀ 10 = 3 − 1 = 2, confirming 10² = 100.

商的对数等于被除数的对数减去除数的对数。例如，log₁₀(1000 ÷ 10) = log₁₀ 1000 − log₁₀ 10 = 3 − 1 = 2，验证了 10² = 100。

Power Law 幂法则

logₐ(Mʳ) = r × logₐ M

The logarithm of a power equals the exponent multiplied by the logarithm of the base. For example, log₃(9⁴) = 4 × log₃ 9 = 4 × 2 = 8.

幂的对数等于指数乘以底数的对数。例如，log₃(9⁴) = 4 × log₃ 9 = 4 × 2 = 8。

3. Change of Base Formula | 换底公式

In many IB problems, you need to evaluate logarithms with bases that are not 10 or e. The change of base formula allows you to convert any logarithm to a convenient base:

在许多IB题目中，需要计算底数不是 10 或 e 的对数。换底公式允许我们将任意对数转换为便捷的底数：

logₐ b = log꜀ b ÷ log꜀ a = (ln b) ÷ (ln a)

In practice, IB students most often choose base 10 or base e, since these are directly available on calculators. For example:

在实际操作中，IB学生最常选择以 10 或 e 为底，因为计算器可以直接计算这两种对数。例如：

log₂ 5 = ln 5 ÷ ln 2 ≈ 1.609 ÷ 0.693 ≈ 2.322

The change of base formula also establishes a useful symmetry: logₐ b × log_b a = 1, since log_b a = ln a ÷ ln b.

换底公式还揭示了一个有用的对称性：logₐ b × log_b a = 1，因为 log_b a = ln a ÷ ln b。

4. Additional Useful Properties | 其他常用性质

Beyond the three core laws, several immediate consequences frequently appear in IB examinations:

除三大核心法则外，以下几个直接推论在IB考试中频繁出现：

logₐ 1 = 0 for any valid base a, because a⁰ = 1.
logₐ 1 = 0 对任何合法底数 a 成立，因为 a⁰ = 1。
logₐ a = 1, because a¹ = a.
logₐ a = 1，因为 a¹ = a。
a^(logₐ x) = x, known as the cancellation identity — exponential and logarithmic functions are inverse operations.
a^(logₐ x) = x，称为消去恒等式——指数函数与对数函数互为逆运算。
If logₐ M = logₐ N, then M = N, provided all logarithms are defined. This is the key to solving logarithmic equations.
若 logₐ M = logₐ N，则 M = N，前提是所有对数均有定义。这是解对数方程的关键。
logₐ (1 ÷ x) = −logₐ x, a direct consequence of the quotient law.
logₐ (1 ÷ x) = −logₐ x，这是商法则的直接推论。

5. The Logarithmic Function | 对数函数

For a fixed base a with a > 0 and a ≠ 1, the logarithmic function is defined by f(x) = logₐ x. Its domain is all positive real numbers (x > 0), and its range is all real numbers.

对于固定的底数 a（a > 0 且 a ≠ 1），对数函数定义为 f(x) = logₐ x。其定义域为所有正实数（x > 0），值域为所有实数。

Key features of the graph of y = logₐ x:

y = logₐ x 的图像关键特征：

The graph passes through the point (1, 0), since logₐ 1 = 0. 图像经过点 (1, 0)，因为 logₐ 1 = 0。
The y-axis (x = 0) is a vertical asymptote; as x → 0⁺, f(x) → −∞. y轴（x = 0）是垂直渐近线；当 x → 0⁺ 时，f(x) → −∞。
If a > 1, the function is strictly increasing; if 0 < a < 1, it is strictly decreasing. 若 a > 1，函数严格递增；若 0 < a < 1，函数严格递减。
The graph is always concave down for a > 1 and concave up for 0 < a < 1. 当 a > 1 时图像始终上凸（凹向下），当 0 < a < 1 时凹向上。

Property \| 性质	a > 1	0 < a < 1
Domain 定义域	x > 0	x > 0
Range 值域	All real numbers 全体实数	All real numbers 全体实数
Monotonicity 单调性	Increasing 递增	Decreasing 递减
x-intercept x轴截距	(1, 0)	(1, 0)
Asymptote 渐近线	x = 0	x = 0

6. Inverse Relationship with Exponential Functions | 与指数函数的反函数关系

The logarithmic function y = logₐ x is the inverse of the exponential function y = aˣ. This inverse relationship manifests itself in three important ways:

对数函数 y = logₐ x 是指数函数 y = aˣ 的反函数。这种反函数关系体现在三个重要方面：

Their graphs are reflections of each other across the line y = x.
它们的图像关于直线 y = x 互为镜像。
Composing them in either order cancels out: logₐ(aˣ) = x and a^(logₐ x) = x.
以任意顺序复合后相互抵消：logₐ(aˣ) = x 且 a^(logₐ x) = x。
If (p, q) lies on y = aˣ, then (q, p) lies on y = logₐ x.
若 (p, q) 位于 y = aˣ 上，则 (q, p) 位于 y = logₐ x 上。

Understanding this relationship is particularly valuable when sketching graphs, solving equations, and evaluating limits in later calculus topics.

理解这层关系，对于后续微积分中绘制图像、求解方程以及计算极限都特别有价值。

7. Solving Logarithmic Equations | 解对数方程

Logarithmic equations appear in every IB paper. The standard strategy involves three steps:

对数方程出现在IB每份试卷中。标准解题策略包含三步：

Step 1 | 第一步: Use logarithmic laws to combine or simplify terms into a single logarithm on each side.

第一步：运用对数法则将各项合并或化简，使等号两边各自成为一个单一对数。

Step 2 | 第二步: Either apply the “logₐ M = logₐ N ⇒ M = N” principle, or convert the equation to exponential form.

第二步：要么运用”logₐ M = logₐ N ⇒ M = N”的消去原则，要么将方程转化为指数形式。

Step 3 | 第三步: Solve the resulting algebraic equation and verify all solutions within the original domain.

第三步：解所得代数方程，并验证所有解满足原方程的定义域。

Worked example 例题:

Solve: log₂ x + log₂(x − 2) = 3

Using the product law: log₂[x(x − 2)] = 3. Converting to exponential form:

利用乘积法则：log₂[x(x − 2)] = 3。转化为指数形式：

x(x − 2) = 2³ = 8

Expanding: x² − 2x − 8 = 0, hence (x − 4)(x + 2) = 0, giving x = 4 or x = −2. Since the domain requires x > 2, the only valid solution is x = 4.

展开：x² − 2x − 8 = 0，因此 (x − 4)(x + 2) = 0，得到 x = 4 或 x = −2。由于定义域要求 x > 2，唯一有效解为 x = 4。

8. The Natural Logarithm and e | 自然对数与 e

The number e ≈ 2.71828 is one of the most important constants in mathematics, arising naturally in compound interest, population growth, and calculus. The natural logarithm ln x = logₑ x is its corresponding logarithmic function.

自然常数 e ≈ 2.71828 是数学中最重要的常数之一，自然出现在复利、人口增长和微分学中。自然对数 ln x = logₑ x 是与之对应的对数函数。

Why is e so special? Because the exponential function f(x) = eˣ has the remarkable property that its derivative is itself: d/dx(eˣ) = eˣ. Correspondingly, the derivative of the natural logarithm is:

为什么 e 如此特殊？因为指数函数 f(x) = eˣ 具有一个重要性质：其导数等于自身，即 d/dx(eˣ) = eˣ。相应地，自然对数的导数为：

d/dx(ln x) = 1/x, for x > 0

This is the only logarithm for which this elegant derivative formula holds without an extra scaling factor. In IB, you will use ln extensively when differentiating and integrating exponential and logarithmic expressions.

这是唯一无需额外缩放因子即具有这种优雅导数公式的对数。在IB中，当你对指数和对数表达式求导与积分时，会大量使用 ln。

9. Logarithmic Equations with Different Bases | 不同底数的对数方程

When solving equations with logarithms of different bases, the change of base formula is indispensable. Consider the equation:

当方程中出现不同底数的对数时，换底公式不可或缺。考虑方程：

log₂ x + log₄ x = 6

Rewrite log₄ x using the change of base formula:

利用换底公式改写 log₄ x：

log₄ x = log₂ x ÷ log₂ 4 = log₂ x ÷ 2

Thus the equation becomes log₂ x + (log₂ x) ÷ 2 = 6, i.e. (3 ÷ 2) × log₂ x = 6, giving log₂ x = 4, so x = 2⁴ = 16.

于是原方程化为 log₂ x + (log₂ x) ÷ 2 = 6，即 (3 ÷ 2) × log₂ x = 6，得 log₂ x = 4，所以 x = 2⁴ = 16。

This example illustrates a general strategy: choose the base that appears most frequently, convert all other logarithms to that base, and reduce the equation to algebraic form.

此例展示了一个通用策略：选择出现最频繁的底数，将其他所有对数转换到该底数，然后化简为代数方程。

10. Transformations of Logarithmic Graphs | 对数图像的变换

In IB exams, you are often asked to sketch or interpret graphs of transformed logarithmic functions. The standard transformation rules apply:

IB考试常要求你绘制或解读经过变换的对数函数图像。标准变换规则适用：

Vertical shift 垂直平移: y = logₐ x + c moves the graph up (c > 0) or down (c < 0).
垂直平移：y = logₐ x + c 使图像向上（c > 0）或向下（c < 0）移动。
Horizontal shift 水平平移: y = logₐ(x − h) moves the graph right (h > 0) or left (h < 0), shifting the vertical asymptote to x = h.
水平平移：y = logₐ(x − h) 使图像向右（h > 0）或向左（h < 0）移动，垂直渐近线也随之移至 x = h。
Vertical stretch 垂直伸缩: y = k·logₐ x stretches the graph vertically by a factor of k.
垂直伸缩：y = k·logₐ x 将图像垂直拉伸 k 倍。
Reflection 反射: y = −logₐ x reflects the graph across the x-axis; y = logₐ(−x) reflects across the y-axis, with domain x < 0.
反射：y = −logₐ x 将图像关于x轴对称；y = logₐ(−x) 将图像关于y轴对称，定义域为 x < 0。

When sketching, always begin with the key point (1, 0) and the vertical asymptote, then apply transformations to both. This disciplined approach minimises errors.

绘图时，先从关键点 (1, 0) 和垂直渐近线出发，再对两者应用变换。这种有条不紊的方法能最大限度地减少错误。

11. Real-World Applications | 实际应用

Logarithms are not merely abstract mathematical objects; they appear throughout science and everyday life. In the IB syllabus, you should be familiar with the most prominent applications.

对数不仅仅是抽象的数学对象，它们广泛存在于科学和日常生活中。在IB教学大纲中，你应该熟悉以下最主要的应用。

Chemistry 化学: pH = −log₁₀[H⁺], where [H⁺] is the hydrogen ion concentration in moles per litre. Each pH unit represents a tenfold change in acidity.

化学：pH = −log₁₀[H⁺]，其中 [H⁺] 是氢离子浓度（单位：摩尔/升）。每个pH单位代表酸度十倍的改变。

Seismology 地震学: The Richter magnitude M = log₁₀(A ÷ A₀), comparing the amplitude of earthquake waves to a reference amplitude. An earthquake of magnitude 6 releases about 32 times more energy than one of magnitude 5.

地震学：里氏震级 M = log₁₀(A ÷ A₀)，将地震波振幅与参考振幅比较。6级地震释放的能量约为5级地震的32倍。

Acoustics 声学: Sound intensity in decibels is calculated as dB = 10 × log₁₀(I ÷ I₀), where I₀ is the threshold of human hearing.

声学：声强以分贝计算，dB = 10 × log₁₀(I ÷ I₀)，其中 I₀ 是人耳听觉阈值。

Exponential models 指数模型: Logarithms are used to linearise exponential growth data, enabling linear regression analysis on semi-log plots — a technique encountered in IB Biology and Economics internal assessments.

指数模型：对数用于将指数增长数据线性化，从而在半对数坐标图上进行线性回归分析——这是IB生物和经济学内部评估中常用的技术。

12. Common Pitfalls and Exam Advice | 常见错误与备考建议

Having taught IB students for many years, I have identified the most frequent errors involving logarithms. Watch for these carefully:

多年辅导IB学生的经验中，我总结出对数板块最常见的错误。请特别留意：

Incorrect domain 忽略定义域: Never forget that logₐ x is only defined for x > 0. Always check solutions against the original domain.
错误一：忽略定义域：切勿忘记 logₐ x 仅对 x > 0 有定义。务必用原方程定义域检验解。
Invalid expansion 错误拆分: logₐ(M + N) ≠ logₐ M + logₐ N. The product law applies only to multiplication, not addition.
错误二：错误拆分：logₐ(M + N) ≠ logₐ M + logₐ N。乘积法则只适用于乘法，而非加法。
Forgetting the base 遗漏底数: When cancelling logarithms, ensure both sides have the same base before applying “one-to-one” reasoning.
错误三：遗漏底数：在消去对数时，务必确保等式两边底数相同，再使用”一一对应”原理。
Dropping the exponent 丢失指数: When using the power law, remember logₐ(xʳ) = r·logₐ x, and this also works in reverse: r·logₐ x = logₐ(xʳ).
错误四：丢失指数：使用幂法则时，牢记 logₐ(xʳ) = r·logₐ x，该式同样可以反用：r·logₐ x = logₐ(xʳ)。

Finally, when revising, practise converting between logarithmic and exponential statements fluently. Try to express every exponential equation you meet in logarithmic form and vice versa. This fluency is the single greatest predictor of success in log-related exam questions.

最后，复习时务必熟练地在对数式与指数式之间进行转换。尝试将你遇到的每一个指数方程改写为对数形式，反之亦然。这种熟练度是考试中解答对数相关问题取得成功的最大保障。

Published by TutorHao | IB Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

Laws of Logarithms and Logarithmic Functions | 对数运算与对数函数