Logarithmic Functions and Their Properties | 对数函数及其性质

📚 Logarithmic Functions and Their Properties | 对数函数及其性质

In the IB Mathematics curriculum, logarithmic functions are not just an algebraic curiosity — they are among the most examined topics in both Analysis & Approaches (AA) and Applications & Interpretation (AI). Understanding how logarithms work, how they behave graphically, and how they transform equations is essential for success in Paper 1 (no calculator) and Paper 2/3 (calculator allowed). This guide walks you through every key property you need, with exam-style insight at every step.

在 IB 数学课程中，对数函数不仅是代数中的理论话题，更是分析与处理方法（AA）与应用与解释方法（AI）两门课程中考查频率极高的内容。无论 Paper 1（不可使用计算器）还是 Paper 2/3（允许使用计算器），理解对数的运算规则、图像特征以及方程变换能力，都是取得高分的基础。本指南将带你逐条梳理所有关键性质，并穿插考试中常见的题型思路。

1. What is a Logarithm? | 什么是对数？

A logarithm answers a simple question: to what exponent must a given base be raised in order to produce a certain number? If a > 0, a ≠ 1, and aˣ = N, then x is called the logarithm of N to base a, written as logₐ N = x.

对数回答的其实是一个简单问题：要把一个给定的底数，乘方多少次（即指数为多少）才能得到某个数？如果 a > 0，a ≠ 1，且 aˣ = N，则称 x 为以 a 为底 N 的对数，记作 logₐ N = x。

Read as “log base a of N”. | 读作“以 a 为底 N 的对数”。
The argument N must be strictly positive: N > 0. | 真数 N 必须严格大于 0。
The base a must satisfy a > 0 and a ≠ 1. | 底数 a 必须满足 a > 0 且 a ≠ 1。

2. Exponents and Logarithms in Disguise | 指数与对数的互化

Logarithms and exponents are two sides of the same coin. The exponential equation and the logarithmic equation are interchangeable ways of expressing exactly the same relationship.

对数与指数是同一枚硬币的两面。指数方程与对数方程是表达同一关系的两种等价形式。

aˣ = N ⇔ logₐ N = x

Two golden identities follow directly from this definition. For any permissible base a and any real exponent x:

由这一定义可直接推出两个黄金恒等式。对任意允许的底数 a 和任意实数指数 x：

logₐ (aˣ) = x | logₐ (aˣ) = x
a^(logₐ N) = N, for N > 0 | a^(logₐ N) = N，其中 N > 0

These identities are frequently used in IB exam questions to simplify expressions before solving algebraic equations.

这两条恒等式在 IB 考试中常被用来先化简表达式，再进一步求解代数方程。

3. Three Core Laws of Logarithms | 对数三大定律

The three core laws of logarithms allow you to rewrite products, quotients, and powers in terms of sums and differences. These are essential tools for both solving equations and proving identities.

对数三大定律允许你将乘积、商和幂改写为和与差的形式。它们既是解方程的重要工具，也是证明恒等式的关键。

Product Law | 乘法法则

logₐ(MN) = logₐ M + logₐ N

Quotient Law | 除法法则

logₐ(M/N) = logₐ M − logₐ N

Power Law | 幂法则

logₐ(Mᵏ) = k logₐ M

All of these laws require M > 0 and N > 0. | 所有定律都要求 M > 0 且 N > 0。
The base a must remain consistent throughout. | 底数 a 必须在整个式子中保持一致。
These laws are valid for any real exponent k. | 这些定律对任意实数指数 k 均成立。

4. The Change of Base Formula | 换底公式

Not all bases are available on a calculator. Standard calculators offer base 10 and base e only. The change of base formula solves this problem elegantly.

并非所有底数都能在计算器上直接使用。常规计算器只提供以 10 为底和以 e 为底的对数。换底公式优雅地解决了这一问题。

logₐ b = log_c b / log_c a, for any c > 0, c ≠ 1

A particularly useful special case is obtained when c = b:

当 c = b 时，可以得到一个特别有用的特例：

logₐ b = 1 / log_b a

Use this formula to evaluate logarithms in any base using your calculator. | 利用该公式，你可以用计算器求任意底数的对数。
The formula is also essential when differentiating or integrating logarithmic functions in IB AA HL. | 在 IB AA HL 中对数函数求导或积分时，该公式同样必不可少。

5. The Logarithmic Function and Its Graph | 对数函数及其图像

The logarithmic function with base a is defined by f(x) = logₐ x, where a > 0 and a ≠ 1. Its graph has a distinctive shape that reflects its inverse relationship with the exponential function.

底数为 a 的对数函数定义为 f(x) = logₐ x，其中 a > 0 且 a ≠ 1。它的图像具有独特的形状，反映了它与指数函数互为反函数的关系。

The graph always passes through (1, 0) and (a, 1). | 图像始终经过点 (1, 0) 和 (a, 1)。
When a > 1, the function is strictly increasing. | 当 a > 1 时，函数严格递增。
When 0 < a < 1, the function is strictly decreasing. | 当 0 < a < 1 时，函数严格递减。
The y-axis, x = 0, is a vertical asymptote. | y 轴，即 x = 0，是函数的垂直渐近线。
The graph of y = logₐ x is the reflection of y = aˣ across the line y = x. | y = logₐ x 的图像是 y = aˣ 的图像关于直线 y = x 反射所得。

6. Domain, Range and Asymptotes | 定义域、值域与渐近线

For the basic function f(x) = logₐ x, the domain is (0, ∞), the range is (−∞, ∞), and the vertical asymptote is x = 0. However, transformed logarithmic functions require extra care.

对基本函数 f(x) = logₐ x，其定义域为 (0, ∞)，值域为 (−∞, ∞)，垂直渐近线为 x = 0。但是，经过变换的对数函数需要格外小心。

For f(x) = logₐ(x − h) + k, the vertical asymptote shifts to x = h, and the domain becomes (h, ∞).

对于 f(x) = logₐ(x − h) + k，垂直渐近线移动到 x = h，定义域变为 (h, ∞)。

The value h represents a horizontal translation. | 常数 h 表示水平平移。
The value k represents a vertical translation. | 常数 k 表示垂直平移。
Reflections such as f(x) = −logₐ x or f(x) = logₐ(−x) flip the graph across an axis and change the domain accordingly. | 反射变换如 f(x) = −logₐ x 或 f(x) = logₐ(−x) 会使图像绕坐标轴翻转，并相应改变定义域。

In IB exams, always state the domain before solving logarithmic equations or inequalities. Many lost marks come from forgetting to check domain restrictions.

在 IB 考试中，求解对数方程或不不等式之前，务必先写出定义域。很多失分都源于忘记检查定义域限制。

7. Natural Logarithm and Common Logarithm | 自然对数与常用对数

Two logarithmic bases appear so often that they have their own notation. The common logarithm uses base 10 and is written as log x. The natural logarithm uses base e, where e ≈ 2.71828, and is written as ln x.

有两个对数底数出现得过于频繁，因此拥有专门记号。常用对数以 10 为底，记作 log x；自然对数以 e 为底，其中 e ≈ 2.71828，记作 ln x。

log x means log₁₀ x. | log x 表示 log₁₀ x。
ln x means log_e x. | ln x 表示 log_e x。
ln e = 1 and ln 1 = 0. | ln e = 1，ln 1 = 0。
The natural logarithm is the inverse function of the exponential function eˣ. | 自然对数是指数函数 eˣ 的反函数。

In IB Mathematics AA, the natural logarithm is especially important because d/dx [ln x] = 1/x, a result that underpins many integration techniques.

在 IB 数学 AA 中，自然对数尤其重要，因为 d/dx [ln x] = 1/x，这一结果支撑着许多积分技巧。

8. Solving Logarithmic Equations | 解对数方程

Logarithmic equations require a systematic approach. In IB exams, you must show clear working and always verify that your final answers lie in the domain.

对数方程需要系统的解题方法。在 IB 考试中，你必须展示清晰的步骤，并始终验证最终答案是否在定义域内。

Example | 例题

Solve log₂(x − 1) + log₂(x + 1) = 3.

解方程 log₂(x − 1) + log₂(x + 1) = 3。

Step 1: Combine the logarithms using the product law. | 步骤 1：用乘法法则合并对数。

log₂[(x − 1)(x + 1)] = 3

Step 2: Convert to exponential form. | 步骤 2：转化为指数形式。

(x − 1)(x + 1) = 2³ = 8

Step 3: Solve the quadratic equation. | 步骤 3：解二次方程。

x² − 1 = 8 ⇒ x² = 9 ⇒ x = ±3

Step 4: Verify the domain. | 步骤 4：验证定义域。

For x = 3, both x − 1 = 2 and x + 1 = 4 are positive, so x = 3 is valid. For x = −3, x − 1 = −4 is negative, so it must be rejected. The solution is x = 3.

当 x = 3 时，x − 1 = 2 和 x + 1 = 4 均为正数，因此 x = 3 有效。当 x = −3 时，x − 1 = −4 为负数，故舍去。最终解为 x = 3。

9. Solving Logarithmic Inequalities | 解对数不等式

When solving logarithmic inequalities, the monotonicity of the logarithm must be taken into account. The direction of the inequality changes depending on whether the base is greater than 1 or between 0 and 1.

求解对数不等式时，必须考虑对数函数的单调性。不等号的方向取决于底数是大于 1，还是介于 0 与 1 之间。

If a > 1, then logₐ x is increasing, so the inequality sign is preserved. | 若 a > 1，则 logₐ x 单调递增，不等号方向不变。
If 0 < a < 1, then logₐ x is decreasing, so the inequality sign is reversed. | 若 0 < a < 1，则 logₐ x 单调递减，不等号方向反转。

Example | 例题

Consider log₂ x > 3. Since the base 2 > 1, the inequality becomes x > 2³ = 8. The solution set is (8, ∞).

例如 log₂ x > 3。因为底数 2 > 1，原不等式化为 x > 2³ = 8，解集为 (8, ∞)。

Now consider log_(1/2) x > 3. Since 0 < 1/2 < 1, the function is decreasing, so the sign flips: x < (1/2)³ = 1/8. Combined with the domain x > 0, the solution is (0, 1/8).

再看 log_(1/2) x > 3。因为 0 < 1/2 < 1，函数单调递减，不等号反向：x < (1/2)³ = 1/8。结合定义域 x > 0，解集为 (0, 1/8)。

Always remember to intersect your algebraic solution with the domain of the logarithm.

永远记得将代数解与对数定义域取交集。

10. Logarithms in the Real World | 对数在实际中的应用

Logarithms appear throughout the real world wherever quantities vary over several orders of magnitude. In the IB Applications & Interpretation course, these contexts are particularly emphasized.

在现实世界中，只要某个量的变化跨越多个数量级，就会频繁出现对数。在 IB 应用与解释课程中，这些实际情境尤其受到重视。

Earthquake magnitude: Richter scale M = log₁₀(I/I₀). | 地震震级：里氏震级 M = log₁₀(I/I₀)。
Acidity: pH = −log₁₀[H⁺]. | 酸碱度：pH = −log₁₀[H⁺]。
Sound intensity: decibels = 10 log₁₀(I/I₀). | 声强级：分贝 = 10 log₁₀(I/I₀)。
Radioactive decay and carbon dating: N(t) = N₀e^(−λt). | 放射性衰变与碳定年：N(t) = N₀e^(−λt)。
Continuous compound interest: A = Pe^(rt). | 连续复利：A = Pe^(rt)。

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Logarithmic Functions and Their Properties | 对数函数及其性质