📚 Mixed Exercise 14: Exponentials and Logarithms | 混合练习14：指数与对数
Mixed Exercise 14 in the Edexcel AS Pure Mathematics course pulls together the whole of Chapter 14 on exponentials and logarithms. It mixes graph interpretation, algebraic manipulation, equation solving and contextual modelling, so it is a very common source of exam-style questions.
Edexcel AS 纯数学课程中的混合练习 14 将第 14 章指数与对数的全部内容整合在一起。它混合了图像解读、代数变形、方程求解和情境建模，因此是考试风格题目的常见来源。
1. What Mixed Exercise 14 Covers | 混合练习14涵盖内容
Mixed Exercise 14 covers exponential functions y = ax and y = ex, their inverse relationship with logarithms, the three laws of logarithms, solving exponential and logarithmic equations, and applications such as exponential growth, decay and modelling.
混合练习 14 涵盖指数函数 y = ax 和 y = ex、它们与对数的逆运算关系、对数的三大法则、指数方程和对数方程的求解，以及指数增长、衰减和建模等应用。
The exercise deliberately mixes skills, so a question may ask you to sketch a curve, solve an equation and interpret a model in the same part.
该练习有意混合技能，因此一道题可能要求你在同一部分中画出曲线、解方程并解释模型。
2. Exponential Functions and Their Graphs | 指数函数及其图像
For a > 1, the graph of y = ax passes through (0,1), rises for all real x, and has the x-axis as a horizontal asymptote. For 0 < a < 1, the graph still passes through (0,1) but decreases as x increases.
当 a > 1 时，y = ax 的图像经过 (0,1)，对所有实数 x 单调上升，并以 x 轴为水平渐近线。当 0 < a < 1 时，图像仍经过 (0,1)，但随着 x 增大而下降。
The gradient of y = ax is proportional to the value of the function itself. The special base e gives the derivative d/dx(ex) = ex, which makes e the natural choice for calculus-based models.
y = ax 的梯度与函数值本身成比例。特殊底数 e 使导数 d/dx(ex) = ex，这使得 e 成为基于微积分的模型中的自然选择。
y = ax (a > 1) ⇒ increasing, y-intercept (0,1), asymptote y = 0
3. Logarithms as Inverse Operations | 对数作为逆运算
The statement y = ax is exactly equivalent to loga y = x. This means that taking a logarithm ‘undoes’ an exponential, and raising a base to a power ‘undoes’ a logarithm.
表达式 y = ax 与 loga y = x 完全等价。这意味着取对数可以’撤销’指数运算，而将底数乘方可以’撤销’对数运算。
y = ax ⇔ x = loga y
Two special values appear constantly: loga 1 = 0 because a0 = 1, and loga a = 1 because a1 = a.
两个特殊值经常出现：loga 1 = 0，因为 a0 = 1；loga a = 1，因为 a1 = a。
4. The Three Core Laws of Logarithms | 对数三大核心法则
The three laws simplify products, quotients and powers inside logarithms:
三条法则简化对数内部的乘积、商和幂：
- loga(xy) = loga x + loga y
- log
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