The Logarithmic Function | 对数函数

📚 The Logarithmic Function | 对数函数

The logarithmic function is one of the most important transcendental functions in mathematics, deeply intertwined with exponential growth and decay. In the IB Mathematics curriculum, a solid grasp of logarithms is essential for tackling exponential equations, modelling real-world scenarios, and preparing for calculus. This article presents a thorough exploration of the logarithmic function, including its definition, fundamental properties, graphical behaviour, equation solving techniques, and practical applications.

对数函数是数学中最重要的超越函数之一，与指数增长和衰减紧密相连。在IB数学课程中，牢固掌握对数知识对于解决指数方程、建立真实世界模型以及为微积分做准备都至关重要。本文深入探讨对数函数，涵盖其定义、基本性质、图像表现、方程求解方法以及实际应用。

1. Definition of a Logarithm | 对数的定义

A logarithm answers the question: to what exponent must a given base be raised in order to obtain a specific number? For any base a > 0, a ≠ 1, and any positive number x > 0, the logarithm base a of x, denoted log_a(x), is the unique exponent y that satisfies a^y = x.

对数回答这样一个问题：给定的底数必须乘方多少次才能得到某个特定的数？对于任意底数 a > 0、a ≠ 1 以及任意正数 x > 0，以 a 为底 x 的对数，记作 log_a(x)，就是满足 a^y = x 的唯一指数 y。

log_a(x) = y ⇔ a^y = x

For example, log₂8 = 3 because 2³ = 8. The value x is called the argument of the logarithm; it must always be strictly positive. There is no real logarithm of zero or a negative number.

例如，log₂8 = 3，因为 2³ = 8。数值 x 被称为对数的真数；它必须始终严格为正。零和负数没有实数对数。

---

2. Logarithmic Form vs Exponential Form | 对数形式与指数形式的互化

Switching between logarithmic and exponential forms is a fundamental skill. The equivalence log_a(y) = x ⇔ a^x = y allows us to solve simple logarithmic equations by rewriting them as exponentials. This conversion also reinforces the idea that logarithms are exponents.

在对数形式和指数形式之间相互转换是一项基本技能。log_a(y) = x ⇔ a^x = y 这组等价关系使我们能够通过改写为指数形式来求解简单的对数方程。这种转换也强化了“对数就是指数”这一概念。

Example: solve log₅(x) = 2. Switching to exponential form gives 5² = x, so x = 25. Conversely, to write 3&sup4; = 81 in logarithmic form we obtain log₃81 = 4.

示例：求解 log₅(x) = 2。转化为指数形式得 5² = x，因此 x = 25。反之，将 3&sup4; = 81 写成对数形式，即 log₃81 = 4。

---

3. Properties of Logarithms | 对数的运算性质

Logarithms obey three fundamental properties derived from the laws of exponents:

对数遵守由指数运算法则导出的三条基本性质：

Product Rule: log_a(MN) = log_a(M) + log_a(N). The log of a product equals the sum of the logs.

积法则：log_a(MN) = log_a(M) + log_a(N)。乘积的对数等于各因数对数的和。

Quotient Rule: log_a(M/N) = log_a(M) − log_a(N). The log of a quotient equals the difference of the logs.

商法则：log_a(M/N) = log_a(M) − log_a(N)。商的对数等于分子的对数减去分母的对数。

Power Rule: log_a(M^p) = p·log_a(M). The log of a power is the exponent times the log of the base.

幂法则：log_a(M^p) = p·log_a(M)。幂的对数等于指数乘以底数的对数。

Additionally, two special values are worth memorising: log_a(1) = 0 because a° = 1, and log_a(a) = 1 because a¹ = a. These properties are frequently used to expand or condense logarithmic expressions.

此外，有两个特殊值值得记住：log_a(1) = 0（因为 a° = 1），以及 log_a(a) = 1（因为 a¹ = a）。这些性质常用于展开或合并对数表达式。

---

广告 / Sponsored

4. Change of Base Formula | 换底公式

When we need to evaluate a logarithm with a base that is not directly available on a calculator, we rely on the change of base formula:

当我们需要计算计算器上不能直接使用的底数的对数时，就要用到换底公式：

log_b(a) = log_c(a) / log_c(b)

Here c can be any positive base different from 1; the most convenient choices are c = 10 (common logarithm) or c = e (natural logarithm). For instance, to compute log₂7 we write log₂ ≈ 2.807. This formula also lets us express any logarithmic function in terms of a single base, which is particularly useful in calculus.

这里 c 可以是任何不等于1的正底数；最便捷的选择是 c = 10（常用对数）或 c = e（自然对数）。例如，要计算 log₂7，可以写成 log₂7 = log 7 / log 2 ≈ 0.8451 / 0.3010 ≈ 2.807。这个公式也允许我们将任何对数函数表示为以单一底数表示的形式，这在微积分中尤其有用。

---

5. Natural Logarithms and Common Logarithms | 自然对数与常用对数

Among all possible bases, two are of paramount importance. The natural logarithm, denoted ln x, uses the base e ≈ 2.71828. It emerges naturally in calculus, continuous compound interest, and growth/decay models. The common logarithm, often written simply as log x, has base 10 and is widely used in science (pH, decibels, Richter scale).

在所有可能的底数中，有两种尤为重要。自然对数，记作 ln x，使用的底数是 e ≈ 2.71828。它在微积分、连续复利以及增长/衰减模型中自然出现。常用对数，通常简写为 log x，以10为底，广泛应用于科学领域（pH值、分贝、里氏震级）。

special bases, these logarithms satisfy ln e = 1, ln 1 = 0, and the inverse relationship e^{ln x} = x for x > 0. Similarly, log 10 = 1 and 10^{log x} = x.

由于底数的特殊性，这些对数满足：ln e = 1，ln 1 = 0，以及反函数关系 e^{ln x} = x（x > 0）。类似地，log 10 = 1 且 10^{log x} = x。

---

6. Graphs of Logarithmic Functions | 对数函数的图像

The graph of y = log_a(x) has a distinctive shape. It always passes through the point (1, 0) and possesses a vertical asymptote at x = 0. The domain is strictly x > 0 and the range is all real numbers.

y = log_a(x) 的图像具有独特的形状。它总是经过点 (1, 0)，并有一条垂直渐近线 x = 0。定义域严格为 x > 0，值域为全体实数。

When a > 1, the function is strictly increasing. As x → 0+, log_a(x) → −∞; as x → ∞, log_a(x) → ∞. For 0 < a < 1, the function is strictly decreasing, with log_a(x) → ∞ as x → 0+ and log_a(x) → −∞ as x → ∞. The larger the base a > 1, the flatter the graph appears for x > 1.

当 a > 1 时，函数严格递增：当 x → 0+ 时，log_a(x) → −∞；当 x → ∞ 时，log_a(x) → ∞。对于 0 < a < 1，函数严格递减：x → 0+ 时，log_a(x) → ∞；x → ∞ 时，log_a(x) → −∞。底数 a > 1 越大，图像在 x > 1 的区域越平缓。

---

7. Transformations of y = log_ax | y = log_ax 的图像变换

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

更多咨询请联系16621398022（同微信）