Working with Natural Logarithms | 自然对数运算

📚 Working with Natural Logarithms | 自然对数运算

Natural logarithms are a cornerstone of A-Level Mathematics, linking exponential growth, calculus, and algebraic manipulation. Mastery of ln x and its properties is essential for solving equations, differentiating complicated functions, and modelling real-world processes.

自然对数是A-Level数学的基石，将指数增长、微积分和代数运算联系起来。掌握ln x及其性质对于求解方程、微分复杂函数和建模实际过程至关重要。

1. Definition and the Number e | 定义与数字 e

ithm to the base e, where e ≈ 2.71828. The constant e can be defined as the limit: e = lim_n→∞ (1 + 1/n)ⁿ. Equivalently, ln x is the power to which e must be raised to obtain x, so y = ln x ⇔ e^y = x.

自然对数，记作ln x，是以e为底的对数，其中e ≈ 2.71828。常数e可定义为极限：e = lim_n→∞ (1 + 1/n)ⁿ。等价地，ln x是使得e的该次幂等于x的指数，即y = ln x ⇔ e^y = x。

values include ln 1 = 0 and ln e = 1. Negative inputs are not allowed in the real context.

ln x 的定义域为 x > 0，关键值包括 ln 1 = 0 和 ln e = 1。在实数范围内，负数输入是不被允许的。

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2. Basic Properties | 基本性质

The natural logarithm obeys the standard log laws:

• ln(ab) = ln a + ln b
• ln(a/b) = ln a – ln b
• ln(aⁿ) = n ln a

These hold for positive a and b.

自然对数遵循标准对数法则：

• ln(ab) = ln a + ln b
• ln(a/b) = ln a – ln b
• ln(aⁿ) = n ln a

这些法则在 a 和 b 为正数时成立。

Example: Simplify ln √x + ln(x³) – 2 ln x. Write √x = x^½, so ln √x = ½ ln x, then ½ ln x + 3 ln x – 2 ln x = (½ + 3 – 2) ln x = 1.5 ln x or (3/2) ln x.

示例：化简 ln √x + ln(x³) – 2 ln x。将 √x 写成 x^½，所以 ln √x = ½ ln x，然后 ½ ln x + 3 ln x – 2 ln x = (½ + 3 – 2) ln x = 1.5 ln x 或 (3/2) ln x。

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3. The Natural Logarithm Function and Its Graph | 自然对数函数及其图像

ses through (1,0) and (e,1). It is increasing but with decreasing slope; as x → 0⁺, ln x → –∞, and as x → ∞, ln x → ∞ slowly. There is a vertical asymptote at x = 0.

y = ln x 的图像经过点 (1,0) 和 (e,1)。它是递增的，但斜率递减；当 x → 0⁺ 时，ln x → –∞，当 x → ∞ 时，ln x 缓慢趋于 ∞。在 x = 0 处有一条垂直渐近线。

Transformations: y = ln(x – moves to x = a); y = ln(kx) can be rewritten as ln k + ln x, shifting vertically by ln k.

变换：y = ln(x – a) 向右平移 a（渐近线移至 x = a）；y = ln(kx) 可改写为 ln k + ln x，垂直移动 ln k。

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4. Derivative of ln x | ln x 的导数

x is d/dx (ln x) = 1/x, for x > 0. More generally, by the chain rule, d/dx [ln(f(x))] = f '(x) / f(x).

ln x 对 x 的导数为 d/dx (ln x) = 1/x (x > 0)。更一般地，根据链式法则，d/dx [ln(f(x))] = f '(x) / f(x)。

Example: If y = ln(sin x), then dy/dx = (cos x) / sin x = cot x. If y = ln(3x²+1), then dy/dx = (6x) / (3x²+1).

示例：如果 y = ln(sin x)，那么 dy/dx = (cos x) / sin x = cot x。如果 y = ln(3x²+1)，那么 dy/dx = (6x) / (3x²+1)。

The second derivative of ln x is d²/dx² (ln x) = –1/x², confirming the graph is always concave down.

ln x 的二阶导数为 d²/dx² (ln x) = –1/x²，这证实其图像始终是凹向下的。

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5. Integral of 1/x | 1/x 的积分

The indefinite integral of 1/x is ∫ (1/x) dx = ln |x| sult is defined for x < 0. For definite integrals, ensure the interval does not include 0.

1/x 的不定积分为 ∫ (1/x) dx = ln |x| + C。绝对值确保了当 x < 0 时结果有定义。对于定积分，须确保区间不包含 0。

Example: ∫₁² (1/x) dx = [ln|x|]₁² = ln 2 – ln 1 = ln 2. Also, ∫ 1/(ax+b) dx = (1/a) ln |ax+b| + C.

示例：∫₁² (1/x) dx = [ln|x|]₁² = ln 2 – ln 1 = ln 2。此外，∫ 1/(ax+b) dx = (1/a) ln |ax+b| + C。

Do not forget the modulus when integrating functions like 1/(x–3) over intervals that span negative values.

当积分函数形如 1/(x–3) 且区间跨越负值时，切勿忘记加上绝对值。

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6. Solving Equations with ln | 求解含 ln 的方程

Equations often require using e^{ln x} = x. To solve ln(2x+1) = 3, exponentiate both sides: e^ln(2x+1) = e³ → 2x+1 = e³ → x = (e³ – 1)/2. Always check solutions are in the domain of the original ln expressions.

方程通常需要利用 e^{ln x} = x。例如求解 ln(2x+1) = 3，对两边取指数：e^ln(2x+1) = e³ → 2x+1 = e³ → x = (e³ – 1)/2。务必检查解是否在原始 ln 表达式的定义域内。

For ln(x+1) + ln(x–2) = ln 3, combine: ln[(x+1)(x–2)] = ln 3, so (x+1)(x–2)=3 → x² – x – 2 = 3 → x² – x – 5 = 0. The roots are x = (1 ± √21)/2. Domain requires x+1>0 and x–2>0 ⇒ x>2, so the only valid solution is x = (1 + √21)/2.

对于 ln(x+1) + ln(x–2) = ln 3，合并：ln[(x+1)(x–2)] = ln 3，因此 (x+1)(x–2)=3 → x² – x – 2 = 3 → x² – x – 5 = 0。根为 x = (1 ± √21)/2。定义域要求 x+1>0 且 x–2>0 ⇒ x>2，所以唯一有效解为 x = (1 + √21)/2。

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7. Using Exponentials to Eliminate ln | 利用指数消除 ln

Any equation of the form ln A = B can be rewritten as A = e^B. This is fundamental when solving for the variable inside a logarithm. Similarly, to eliminate an exponential, take natural logs: e^kx = C ⇒ kx = ln C, provided C>0.

任何形如 ln A = B 的方程都可以改写为 A = e^B。这是在求解对数内的变量时的基本操作。同样，为了消除指数，可对两边取自然对数：e^kx = C ⇒ kx = ln C，前提是 C>0。

Solve 5e^2x = 20: divide by 5 → e^2x = 4 → 2x = ln 4 → x = ln 4 / 2 = ln(4^½) = ln 2.

求解 5e^2x = 20：除以 5 → e^2x = 4 → 2x = ln 4 → x = ln 4 / 2 = ln(4^½) = ln 2。

When a constant appears, such as 2, rewrite as ln(e²) to combine with other logarithmic terms.

当出现常数如 2 时，可将其重写为 ln(e²) 以便与其他对数项合并。

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8. Logarithmic Differentiation | 对数微分法

When a function involves a variable raised to a variable power, e.g., y = x^x both sides: ln y = x ln x. Differentiate implicitly: (1/y) dy/dx = ln x + 1, so dy/dx = x^x(ln x + 1). This technique also simplifies products and quotients.

当函数涉及变量作为指数，例如 y = x^x，对两边取自然对数：ln y = x ln x。隐式微分得：(1/y) dy/dx = ln x + 1，因此 dy/dx = x^x(ln x + 1)。该技巧也可用于简化乘积和商的导数。

For y = (x²√x+1)/(x–3)³, take ln: ln y = 2 ln x + ½ ln(x+1) – 3 ln(x–3). Then dy/dx = y [

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