Working with Natural Logarithms | 自然对数运算

📚 Working with Natural Logarithms | 自然对数运算

Natural logarithms are logarithms with base e, where e ≈ 2.71828. They appear throughout A-Level calculus, exponential models, and algebraic equations, so a confident command of ln x is essential for Edexcel exam success.

自然对数是以 e 为底的对数，e ≈ 2.71828。它在 A-Level 的微积分、指数模型和代数方程中无处不在，因此熟练掌握 ln x 是 Edexcel 考试取得高分的关键。

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1. Definition of ln x | ln x 的定义

If y = ln x, then by definition x = eʸ. The natural logarithm is the inverse function of the exponential function eˣ.

若 y = ln x，则按定义 x = eʸ。自然对数是指数函数 eˣ 的反函数。

Two key values follow immediately: ln 1 = 0 because e⁰ = 1, and ln e = 1 because e¹ = e.

两个关键值可直接得出：ln 1 = 0，因为 e⁰ = 1；ln e = 1，因为 e¹ = e。

The notation ln x means logₑ x, so the base does not need to be written explicitly.

记号 ln x 表示 logₑ x，因此底数无需显式写出。

The equivalence eʸ = x ⇔ y = ln x is the foundation for solving almost every natural logarithmic equation.

等价关系 eʸ = x ⇔ y = ln x 是求解几乎所有自然对数方程的基础。

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2. Graph and Domain of y = ln x | y = ln x 的图像与定义域

The function y = ln x is defined only for x > 0. Its range is the set of all real numbers.

函数 y = ln x 只在 x > 0 时有定义，值域为全体实数。

The graph passes through (1, 0), increases for all x > 0, and has a vertical asymptote at x = 0.

图像经过点 (1, 0)，在 x > 0 上始终递增，并以 x = 0 为竖直渐近线。

As x approaches 0 from the right, ln x tends to −∞. As x increases, ln x grows without bound but very slowly.

当 x 从右侧趋近于 0 时，ln x 趋于 −∞；当 x 增大时，ln x 无限增大但增长非常缓慢。

Because ln x is the inverse of eˣ, the graph of y = ln x is the reflection of y = eˣ in the line y = x.

由于 ln x 是 eˣ 的反函数，y = ln x 的图像是 y = eˣ 关于直线 y = x 的对称图形。

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3. Laws of Natural Logarithms | 自然对数的运算律

Natural logarithms obey the same laws as logarithms to any base. These laws are valid when the arguments are positive.

自然对数遵循与任意底对数相同的运算律。当真数为正时，这些法则成立。

ln(ab) = ln a + ln b

The product rule states that the logarithm of a product is the sum of the logarithms.

乘法法则指出，乘积的对数等于对数之和。

ln(a/b) = ln a − ln b

The quotient rule states that the logarithm of a quotient is the difference of the logarithms.

除法法则指出，商的对数等于对数之差。

ln(aⁿ) = n ln a

The power rule states that the logarithm of a power is the exponent times the logarithm of the base. This holds for real n.

幂法则指出，幂的对数等于指数乘以底数的对数。这对实数 n 成立。

In particular, ln(1/x) = −ln x and ln(√x) = ½ ln x.

特别地，ln(1/x) = −ln x，ln(√x) = ½ ln x。

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4. Solving Equations with ln x | 解含 ln x 的方程

To solve an equation such as ln(2x − 3) = 1, rewrite it in exponential form: 2x − 3 = e¹.

解方程 ln(2x − 3) = 1 时，先改写为指数形式：2x − 3 = e¹。

Then x = (e + 3)/2. Always check that the argument of each logarithm is positive.

然后 x = (e + 3)/2。务必检验每个对数的真数为正。

If ln A = ln B, then A = B provided A > 0 and B > 0.

若 ln A = ln B，则在 A > 0 且 B > 0 的条件下 A = B。

Example: ln(x + 1) + ln(x − 1) = ln 8. Combine to ln[(x + 1)(x − 1)] = ln 8, so x² − 1 = 8, giving x = ±3. Since x > 1 is required, only x = 3 is valid.

例如：ln(x + 1) + ln(x − 1) = ln 8。合并得 ln[(x + 1)(x − 1)] = ln 8，因此 x² − 1 = 8，解得 x = ±3。由于需要 x > 1，仅取 x = 3。

When combining logarithmic terms, always re-state the domain before cancelling logarithms.

合并对数项时，务必在消去对数前重新写出定义域。

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5. Equations with eˣ and ln x | 含 eˣ 与 ln x 的方程

The functions eˣ and ln x are inverses, so e^(ln x) = x for x > 0 and ln(eˣ) = x for all real x.

函数 eˣ 与 ln x 互为反函数，

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