Exponential Functions (IB Math) | 指数函数专题精讲

📚 Exponential Functions (IB Math) | 指数函数专题精讲

An exponential function is a function of the form f(x) = a^x, where a is a positive real number not equal to 1. This function is defined for every real value of x. The behaviour of its graph changes dramatically depending on whether a > 1 or 0 < a < 1.

指数函数是形如 f(x) = a^x 的函数，其中 a 是正实数且不等于 1。该函数对一切实数 x 都有定义。它的图像行为会根据 a > 1 还是 0 < a < 1 而发生显著变化。

1. What is an Exponential Function? | 什么是指数函数

In IB Mathematics, the exponential function is defined as f(x) = a^x with base a > 0 and a ≠ 1. Its domain is ℝ, the set of all real numbers; its range is the set of positive real numbers (0, ∞). The graph always passes through the point (0, 1) because a⁰ = 1. There is no real x-intercept, and the x-axis (y = 0) is a horizontal asymptote.

在 IB 数学中，指数函数定义为 f(x) = a^x，底数 a > 0 且 a ≠ 1。其定义域为全体实数 ℝ，值域为所有正实数 (0, ∞)。图像恒经过点 (0, 1)，因为 a⁰ = 1。函数没有实数 x 截距，x 轴（即 y = 0）是水平渐近线。

2. Properties of the Graph | 指数函数的图像特征

If a > 1, the graph rises to the right and approaches the x-axis as x → −∞. If 0 < a < 1, the graph falls to the right and still approaches the x-axis as x → +∞. The graph is always above the x-axis and is strictly monotonic, so it passes the horizontal line test. Transformations such as y = a^x−h + k shift the graph, and the horizontal asymptote becomes y = k.

当 a > 1 时，图像向右上升，而当 x → −∞ 时向 x 轴无限靠近。当 0 < a < 1 时，图像向右下降，当 x → +∞ 时向 x 轴无限靠近。图像始终位于 x 轴上方，且严格单调，因此能通过水平线检验。平移变换如 y = a^x−h + k 会移动图像，水平渐近线变为 y = k。

3. The Laws of Exponents | 指数运算法则

These laws are essential for simplifying exponential expressions and solving equations. They hold for all real values of m and n, provided the bases are positive.

这些法则对于化简指数表达式和求解方程至关重要。只要底数为正，它们对一切实数 m 和 n 都成立。

Law 法则	Example 示例
a^m × aⁿ = a^m+n	2³ × 2⁴ = 2⁷
a^m / aⁿ = a^m−n	5⁶ / 5² = 5⁴
(a^m)ⁿ = a^mn	(3²)⁵ = 3¹⁰
(ab)ⁿ = aⁿbⁿ	(2x)⁴ = 16x⁴
(a/b)ⁿ = aⁿ / bⁿ	(7/3)² = 49/9
a⁰ = 1, a⁻ⁿ = 1 / aⁿ	10⁰ = 1, 2⁻³ = 1/8
a^1/n = ⁿ√a	8^1/3 = ∛8 = 2

4. Exponential Equations | 指数方程

To solve an exponential equation, first try to write both sides with the same base. If a^u = a^v, then u = v. When the bases cannot be made equal, take logarithms of both sides.,

解指数方程时，首先尝试将两边写成同底。若 a^u = a^v，则 u = v。当两边无法化成同底时，可对两边取对数。

Example 1: 2^3x−1 = 16

Since 16 = 2⁴, we have 3x − 1 = 4, hence x = 5/3.

因为 16 = 2⁴，所以 3x − 1 = 4，从而 x = 5/3。

Example 2: 3^x+2 = 7

Taking natural logs: (x + 2) ln 3 = ln 7, so x = ln 7 / ln 3 − 2.

取自然对数得 (x + 2) ln 3 = ln 7，因此 x = ln 7 / ln 3 − 2。

5. Exponential Inequalities | 指数不等式

When solving exponential inequalities, the monotonicity of the base is essential. If a > 1, the inequality direction is preserved. If 0 < a < 1, the direction is reversed.

解指数不等式时，底数的单调性至关重要。当 a > 1 时，不等号方向不变；当 0 < a < 1 时，不等号方向需要颠倒。

Example 1: 2^x > 8

8 = 2³, base 2 > 1, so x > 3.

8 = 2³，底数 2 > 1，所以 x > 3。

Example 2: (1/2)^x > 4

(1/2)^x > (1/2)⁻². Since 0 < 1/2 < 1, the direction reverses, giving x < −2.

(1/2)^x > (1/2)⁻²。因为 0 < 1/2 < 1，不等号方向反转，得到 x < −2。

6. The Natural Exponential and e | 自然指数与 e

The irrational number e ≈ 2.71828 is the base of natural logarithms. The function f(x) = e^x is called the natural exponential function. It appears naturally in calculus and in real-world growth processes. For continuous compound interest, the amount is given by A = Pe^rt.

无理数 e ≈ 2.71828 是自然对数的底数。函数 f(x) = e^x 称为自然指数函数。它在微积分和现实增长过程中自然出现。连续复利中，总额由 A = Pe^rt 给出。

7. Applications: Growth and Decay | 应用：增长与衰减

Exponential growth is modelled by N(t) = N₀e^kt with k > 0. Exponential decay corresponds to k < 0. A common decay model is half-life: N(t) = N₀(1/2)^t/h, where h is the half-life. These models are used in biology, chemistry, economics and physics.

指数增长由 N(t) = N₀e^kt 建模，其中 k > 0。指数衰减则对应 k < 0。常见的衰减模型是半衰期：N(t) = N₀(1/2)^t/h，其中 h 是半衰期。这些模型广泛应用于生物、化学、经济和物理。

Example: A radioactive substance decays from 100 g to 25 g in 12 days. Find its half-life.

Using 25 = 100(1/2)^12/h, we get 1/4 = (1/2)^12/h. Since 1/4 = (1/2)², 12/h = 2, so h = 6 days.

由 25 = 100(1/2)^12/h，得 1/4 = (1/2)^12/h。因为 1/4 = (1/2)²，所以 12/h = 2，即 h = 6 天。

8. Relation to Logarithms | 与对数函数的关系

The exponential function y = a^x and the logarithmic function y = log_ax are inverses of each other. Therefore, for a > 0 and a ≠ 1:

指数函数 y = a^x 与对数函数 y = log_ax 互为反函数。因此，对于 a > 0 且 a ≠ 1：

y = a^x ⇔ x = log_ay

The natural logarithm is the inverse of e^x: ln x = log_ex. The change of base rule is log_ab = ln b / ln a.

自然对数是 e^x 的反函数：ln x = log_ex。换底公式为 log_ab = ln b / ln a。

9. Differentiation and Integration | 导数和积分

In IB calculus, the derivative of a general exponential function is:

在 IB 微积分中，一般指数函数的导数为：

d/dx (a^x) = a^x ln a

In particular, d/dx (e^x) = e^x. Using the chain rule, if u = u(x), then d/dx (a^u) = a^u ln a · u′.

特别地，d/dx (e^x) = e^x。根据链式法则，若 u = u(x)，则 d/dx (a^u) = a^u ln a · u′。

For integration:

积分方面：

∫ a^x dx = a^x / ln a + C (a ≠ 1) 以及 ∫ e^x dx = e^x + C

10. Common Pitfalls and Exam Tips | 常见错误与考试要点

Students often confuse −a^x with (−a)^x. Remember that −a^x means −(a^x). Another common error is assuming (a + b)ⁿ = aⁿ + bⁿ, which is false. When solving inequalities, pay close attention to the base: for 0 < a < 1, the inequality sign must flip.

学生常混淆 −a^x 与 (−a)^x。注意 −a^x 表示 −(a^x)。另一个常见错误是认为 (a + b)ⁿ = aⁿ + bⁿ，这是错误的。解不等式时要特别注意底数：当 0 < a < 1 时，必须反转不等号方向。

Exam tips: always state the domain and range when sketching; use the same base for quick solutions; take logarithms when the variable is in the exponent; and in calculus, remember the factor ln a when differentiating a^x.

考试要点：画图时始终注明定义域和值域；优先化为同底快速求解；当变量位于指数中时取对数；在微积分中，对 a^x 求导时不要遗漏因子 ln a。

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

Exponential Functions (IB Math) | 指数函数专题精讲