📚 D - Natural logarithms | 自然对数
Natural logarithms form a cornerstone of IB Mathematics, appearing across algebra, calculus, and real‑world modelling. Understanding ln x and its relationship with the exponential function ex is essential for success in both Analysis & Approaches and Applications & Interpretation. This article walks you through the definitions, properties, differentiation, integration, and common pitfalls involving natural logs, supported by examples that mirror IB exam style.
自然对数是 IB 数学的重要基石，贯穿代数、微积分和现实建模。理解 ln x 及其与指数函数 ex 的关系，对于在分析与方法和应用与解释两门课程中取得成功至关重要。本文带你梳理自然对数的定义、性质、微分、积分和常见误区，并提供贴近 IB 考试风格的示例。
1. Definition of the Natural Logarithm | 自然对数的定义
of a positive number x, written as ln x, is the logarithm to the base e, where e is Euler's number, approximately 2.718281828. Formally, y = ln x if and only if ey = x. This makes ln x the inverse function of the exponential function ex. The notation 'ln' comes from the Latin logaritmus naturalis.
正数 x 的自然对数记作 ln x，是以 e 为底的对数，其中 e 是欧拉数，约等于 2.718281828。严格来说，y = ln x 当且仅当 ey = x。因此 ln x 是指数函数 ex 的反函数。记号 'ln' 来源于拉丁文 logaritmus naturalis。
In IB Mathematics, you must remember that the argument of a natural log must always be strictly positive: ln x is defined only for x > 0. This domain restriction is frequently tested in equation‑solving and calculus questions.
在 IB 数学中，必须牢记自然对数的真数必须严格为正：ln x 仅在 x > 0 时有定义。这个定义域限制经常在解方程和微积分题目中被考察。
2. The Number e | 数字 e
The constant e is an irrational and transcendental number that can be defined as the limit limn→∞ (1 + 1/n)n. Another definition is the infinite series Σn=0∞ 1/n!. The value e ≈ 2.71828 appears naturally in calculus because the function ex is its own derivative, and the slope of y = ln x at x = 1 is exactly 1.
常数 e 是一个无理且超越的数，可定义为极限 limn→∞ (1 + 1/n)n。另一个定义是无穷级数 Σn=0∞ 1/n!。e ≈ 2.71828 在微积分中自然出现，因为 ex 的导数就是它本身，而 y = ln x 在 x = 1 处的斜率恰好为 1。
A key IB fact: ln e = 1, since e1 = e. Likewise, ln 1 = 0 because e0 = 1. These identities are used repeatedly to simplify expressions.
IB 中的一个关键事实：ln e = 1，因为 e1 = e。同样地，ln 1 = 0，因为 e0 = 1。这些恒等式被反复用于简化表达式。
3. Properties of Natural Logarithms | 自然对数的性质
Natural logarithms obey the same three fundamental laws as any logarithm. These properties are indispensable for expanding, condensing, and solving logarithmic expressions.
自然对数遵循与任何对数相同的三个基本法则。这些性质在展开、合并和解对数表达式时不可或缺。
Product rule: ln(ab) = ln a + ln b, for a, b > 0.
积的法则： ln(ab) = ln a + ln b，其中 a, b > 0。
ln(ab) = ln a + ln b
Quotient rule: ln(a/b) = ln a – ln b, for a, b > 0.
商的法则： ln(a/b) = ln a – ln b，其中 a, b > 0。
ln(a/b) = ln a − ln b
Power rule: ln(an) = n ln a. This is particularly useful when the unknown appears in an exponent.
幂的法则： ln(an) = n ln a。当未知数出现在指数中时，这条性质尤其有用。
ln(an) = n ln a
Many IB problems require you to combine these rules in reverse, e.g. rewriting 2 ln x + 3 ln y as ln(x2y3).
许多 IB 问题要求你逆向组合这些法则，例如将 2 ln x + 3 ln y 改写为 ln(x2y3)。
4. Change of Base Formula | 换底公式
Although your calculator can compute logarithms with any base, the change of base formula is essential for algebraic manipulation. It states that for any positive a, b, c where b ≠ 1 and c ≠ 1:
尽管计算器可以计算任意底数的对数，但换底公式在代数变形中必不可少。对任意正数 a, b, c 且 b ≠ 1, c ≠ 1，有：
logb a = (ln a) / (ln b) = (logc a) / (logc b)
In IB exams, you rarely need a base other than 10 or e, but the formula allows you to solve equations like 2x = 32x+1 by taking natural logs of both sides.
在 IB 考试中，你很少需要除 10 或 e 以外的底数，但该公式允许你通过对方程两边取自然对数来求解诸如 2x = 32x+1 的方程。
5. Graphs and Key Features | 图像与主要特征
0. It passes through (1, 0) and increases slowly as x → ∞. The vertical line x = 0 (the y‑axis) is a vertical asymptote; as x → 0+, ln x → –∞. The inverse graph y = ex is the reflection of y = ln x in the line y = x.
y = ln x 的图像仅在 x > 0 上有定义。它经过点 (1, 0)，并随 x → ∞ 缓慢递增。直线 x = 0（y 轴）是垂直渐近线；当 x → 0+ 时，ln x → –∞。反函数 y = ex 的图像是 y = ln x 关于直线 y = x 的反射。
Key features to remember: the x‑intercept is always (1, 0); there is no y‑intercept; the function is concave down (second derivative –1/x2 < 0). These properties help when sketching transformed functions like y = ln(x – 2) + 3.
需要牢记的主要特征：x 截距永远是 (1, 0)；没有 y 截距；函数是凹向下的（二阶导数为 –1/x2 < 0）。这些性质在绘制诸如 y = ln(x – 2) + 3 的变换函数图像时很有帮助。
6. Solving Equations Using Natural Logarithms | 用自然对数解方程
When an unknown is in the exponent, taking natural logs on both sides is often the most efficient method. For instance, to solve e2x = 5, take ln of both sides: 2x = ln 5, so x = (ln 5)/2.
当未知数出现在指数中时，两边取自然对数通常是最有效的方法。例如，解 e2x = 5，两边取 ln 得 2x = ln 5，从而 x = (ln 5)/2。
quation contains logarithms, you may need to combine them first: equation ln(x + 2) + ln(x – 1) = ln 8 can be rewritten as ln[(x + 2)(x – 1)] = ln 8. Then, equating arguments gives (x + 2)(x – uadratic. Always check that solutions satisfy the original domain (x > 1 in this case).
如果方程中含有对数，你可能需要先将它们合并：方程 ln(x + 2) + ln(x – 1) = ln 8 可改写为 ln[(x + 2)(x – 1)] = ln 8。然后令真数相等得 (x + 2)(x – 1) = 8，化为一元二次方程。务必检验解是否满足原始定义域（在此例中 x > 1）。
For exponential functions with different bases, use the power rule: 3x = 2x+1 ⇒ x ln 3 = (x + 1) ln 2, then solve the linear equation.
对于底数不同的指数函数，使用幂法则：3x = 2x+1 ⇒ x ln 3 = (x + 1) ln 2，然后解一次方程。
7. Derivative of ln x and Related Functions | 自然对数函数及其相关函数的导数
The derivative of the natural logarithm is simple yet powerful:
自然对数的导数既简单又强大：
d/dx (ln x) = 1/x, for x > 0
By the chain rule, d/dx [ln(u(x))] = u'(x) / u(x). For example, if y = ln(5x2 + 3), then dy/dx = (10x) / (5x2 + 3). This pattern appears frequently in IB questions, especially when combined with product or quotient rules.
根据链式法则，d/dx [ln(u(x))] = u'(x) / u(x)。例如，若 y = ln(5x2 + 3)，则 dy/dx = (10x) / (5x2 + 3)。这一模式在 IB 试题中频繁出现，尤其与乘法法则或除法法则结合时。
f the argument is an absolute value, the derivative expression can be extended to d/dx (ln |x|) = 1/x for x ≠ 0.
若自变量为绝对值形式，导数公式可推广为 d/dx (ln |x|) = 1/x，其中 x ≠ 0。
8. Integrals Involving 1/x and ln x | 涉及 1/x 和 ln x 的积分
The fundamental integral that every IB student must know is:
每个 IB 学生都必须掌握的基本积分是：
∫ (1/x) dx = ln |x| + C
This arises because the derivative of ln |x| is 1/x. More generally, a recognition of the form f'(x)/f(x) leads to:
这是因为 ln |x| 的导数为 1/x。更一般地，识别 f'(x)/f(x) 的形式可得：
∫ f'(x)/f(x) dx = ln |f(x)| + C
Example: Evaluate ∫ (2x + 1)/(x2 + x) dx. Since the numerator is the derivative of the denominator, the integral becomes ln |x2 + x| + C. In definite integration, remember to apply the absolute value when appropriate, and ensure the interval does not cross any singularities.
示例：计算 ∫ (2x + 1)/(x2 + x) dx。由于分子是分母的导数，积分结果是 ln |x2 + x| + C。在定积分中，记得在适当的时候使用绝对值，并确保积分区间不跨过任何不连续点。
9. Logarithmic Differentiation | 对数微分法
in both the base and the exponent, logarithmic differentiation is a lifesaver. Take y = xsin x implicitly: (1/y) dy/dx = cos x · ln x + sin x · (1/x). Finally, multiply by y: dy/dx = xsin x ( cos x · ln x + (sin x)/x ).
对于变量同时出现在底数和指数中的复杂函数，对数微分法是救星。以 y = xsin x 为例：首先两边取自然对数，ln y = sin x · ln x。然后隐式求导：(1/y) dy/dx = cos x · ln x + sin x · (1/x)。最后乘以 y：dy/dx = xsin x ( cos x · ln x + (sin x)/x )。
hnique also simplifies derivatives of products or quotients with many factors, e.g. y = (x2√x+1)/(x–3)4. Taking ln y converts a messy quotient into a sum of simpler terms that are easy to differentiate.
这一技巧也可简化含有多个因式的乘积或商的导数，例如 y = (x2√x+1)/(x–3)4。对 y 取自然对数可将复杂的商化为简单项的和，从而易于求导。
10. Applications: Exponential Growth and Decay | 应用：指数增长与衰减
are indispensable in modelling continuous growth and decay. The standard model is P(t) = P0 ekt, where P0 owth (k > 0) or decay (k < 0) rate, and t is time. To find the doubling time or half‑life, set P(t) = 2P0 or P(t) = ½P0 and solve for t using natural logs: t = (ln 2) / k for doubling, t = (ln 2) / |k| for half‑life.
自然对数在连续增长和衰减建模中必不可少。标准模型为 P(t) = P0 ekt，其中 P0 是初始量，k 是连续增长（k > 0）或衰减（k < 0）率，t 是时间。要求翻倍时间或半衰期，可设 P(t) = 2P0 或 P(t) = ½P0，然后利用自然对数求解 t：翻倍时间 t = (ln 2) / k，半衰期 t = (ln 2) / |k|。
om given data. For example, if a culture grows from 200 to 350 in 3 hours, use 350 = 200 e3k. Thus e3k = 1.75, so k = (ln 1.75)/3. Always interpret the sign and magnitude of k in context.
IB 题常要求从给定数据求 k。例如，一个培养物在 3 小时内从 200 增长到 350，利用 350 = 200 e3k，得 e3k = 1.75，因此 k = (ln 1.75)/3。注意根据上下文解释 k 的符号和大小。
11. Common Mistakes | 常见错误
Avoid these frequent pitfalls when working with natural logarithms:
使用自然对数时避免以下常见错误：
– ieving ln(a + b) = ln a + ln b. This is false; logarithms do not distribute over addition.
– 误以为 ln(a + b) = ln a + ln b。这是错误的；对数不能对加法分配。
– Forgetting the domain: ln(negative) is undefined. Always check that arguments remain positive when solving equations.
– 忘记定义域：ln(负数) 无定义。解方程时务必检查真数保持为正。
– Applying the power rule incorrectly: ln(2x) is NOT 2 ln x; it is ln 2 + ln x.
– 幂法则应用不当：ln(2x) ≠ 2 ln x，而应为 ln 2 + ln x。
– Dropping the absolute value in ∫ (1/x) dx. The correct antiderivative is ln|x| + C, which ensures validity for negative x.
– 在 ∫ (1/x) dx 中遗漏绝对值。正确的原函数为 ln|x| + C，以确保对负 x 也成立。
– Confusing ln(ex) = x with eln x = x; both are true for suitable domains, but forgetting the domain x>0 for the latter.
– 混淆 ln(ex) = x 与 eln x = x；在适当定义域下两者都成立，但后者要求 x>0，常被遗忘。
12. Exam Tips and Tricks | 考试技巧与
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