📚 Mixed Exercise 14 | 混合练习 14
This article provides a comprehensive walkthrough of the key concepts and problem types found in Mixed Exercise 14 from the Edexcel A Level Mathematics Pure Year 1 textbook, focusing on exponentials and logarithms. We will revise the core principles, demonstrate essential techniques, and work through representative examples to build your confidence for the exam. Whether you are struggling with exponential equations or applying logs in real-world contexts, this revision guide is designed to support your learning step by step.
本文全面梳理了 Edexcel A Level 数学 Pure Year 1 教材中混合练习 14 的核心概念与典型题型，主题为指数与对数。我们将复习基本原理，展示关键技巧，并通过代表性例题带你一步步建立考试信心。无论你是在指数方程上遇到困难，还是想更好地将对数应用到实际情境中，这篇复习指南都能为你提供循序渐进的支持。
1. The Exponential Function | 指数函数
The exponential function, f(x) = aˣ el, the most important base is e ≈ 2.71828, the irrational constant. The function f(x) = eˣ ly as x increases. For a > 1, the function is increasing; for 0 < a < 1, it is decreasing.
指数函数 f(x) = aˣ（a > 0 且 a ≠ 1）对所有实数 x 有定义。在 A Level 中，最重要的底数是无理常数 e ≈ 2.71828，函数 f(x) = eˣ 称为自然指数函数。它的图像经过 (0, 1)，始终为正，并随 x 增大而快速增长。当 a > 1 时，函数递增；当 0 < a < 1 时，函数递减。
You must be comfortable sketching y = aˣ, y = a⁻ˣ, and transformations like y = aˣ⁺³ − 2. The derivative of eˣ is itself, d/dx(eˣ) = eˣ, and by chain rule, d/dx(eᵏˣ) = k eᵏˣ. This property is foundational for calculus with exponentials.
你需要熟练画出 y = aˣ、y = a⁻ˣ 以及变换后的图像，如 y = aˣ⁺³ − 2。eˣ 的导数还是它本身，即 d/dx(eˣ) = eˣ，运用链式法则可得 d/dx(eᵏˣ) = k eᵏˣ。这一性质是指数函数微积分运算的基础。
o evaluate or sketch exponential functions, or to solve equations by recognising that if aˣ ins many exam-style questions. Always check the domain and range when sketching: for y = 2ˣ, the domain is all real numbers, and the range is y > 0.
在混合练习 14 中，你可能需要计算或绘制指数函数的图像，或利用“若 aˣ = aⁿ 则 x = n”来解方程。这个简单思路是许多考题的基础。绘图时一定要检查定义域和值域：例如 y = 2ˣ 的定义域是所有实数，值域是 y > 0。
2. Introduction to Logarithms | 对数简介
The logarithm is the inverse operation of exponentiation. The statement y = aˣ arithm base 10 is often written as log y, and the natural logarithm, base e, is denoted ln y.
对数是取指数的逆运算。等式 y = aˣ 等价于 x = logₐ y。记号 logₐ y 表示的是为了使 a 变成 y 而必须对 a 取的幂次。例如 log₂ 8 = 3，因为 2³ = 8。以 10 为底的对数常写作 log y，而以 e 为底的自然对数记作 ln y。
know the special values: logₐ 1 = 0 (because a⁰ = 1) and logₐ a = 1 (because a¹ = a). These are frequently used to simplify expressions.
对数仅对正实数有定义，即 logₐ x 只有当 x > 0 时才存在。底数 a 必须为正且不等于 1。你应当知道特殊值：logₐ 1 = 0（因为 a⁰ = 1）和 logₐ a = 1（因为 a¹ = a）。这些特殊值经常被用来化简表达式。
nvert between exponential and logarithmic forms to solve problems. For instance, rewrite 3ˣ n to evaluate log₅ 125. Mastery of this inverse relationship is essential before tackling more complex equations.
在混合练习 14 中，你会通过指数形式与对数形式的相互转换来解决问题。例如，将 3ˣ = 20 改写为 x = log₃ 20，或利用定义来求 log₅ 125 的值。在解决更复杂的方程之前，掌握这一互逆关系至关重要。
3. Laws of Logarithms | 对数运算法则
ndices. Law 1: logₐ (xy) = logₐ x + logₐ y. Law 2: logₐ (x/y) = logₐ x − logₐ y. Law 3: logₐ (xⁿ) = n logₐ x. These hold for any base a > 0, a ≠ 1, and for x, y > 0.
对数的三条基本运算法则让你能够灵活操作和组合对数表达式。它们直接由指数法则推导而来。法则一：logₐ (xy) = logₐ x + logₐ y。法则二：logₐ (x/y) = logₐ x − logₐ y。法则三：logₐ (xⁿ) = n logₐ x。这些法则对所有 a > 0, a ≠ 1 且 x, y > 0 成立。
lculator, typically by converting to base 10 or e. A special case is logₐ b = 1 / log_b a.
你还应了解换底公式：logₐ b = log_c b / log_c a，这在用计算器求对数值时特别有用，通常可以转换为以 10 或 e 为底。一个特例是 logₐ b = 1 / log_b a。
Mixed Exercise 14 questions will require you to apply these laws in both directions: e.g., simplify 2 log x + 3 log y into log (x² y³), or expand log (a²/√b) into 2 log a − ½ log b. Mistakes often arise from forgetting that log (x + y) cannot be simplified using these laws. Practise spotting which expressions can be condensed and which cannot.
混合练习 14 中的问题会要求你双向运用这些法则：例如将 2 log x + 3 log y 化简为 log (x² y³)，或将 log (a²/√b) 展开为 2 log a − ½ log b。常见的错误是误以为 log (x + y) 也能用这些法则简化。要勤于分辨哪些表达式可以压缩，哪些不能。
4. Solving Exponential Equations | 解指数方程
ponents, and (2) taking logarithms on both sides. When the bases can easily be matched, the first method is quicker. For example, solve 2ˣ⁻¹ = 8. Recognise 8 = 2³, so x − 1 = 3, giving x = 4.
指数方程是指未知数出现在指数位置的方程。主要有两种策略：(1) 将两边改写为同底数然后令指数相等；(2) 两边同时取对数。当底数容易匹配时，第一种方法更快。例如解 2ˣ⁻¹ = 8，因 8 = 2³，故 x − 1 = 3，解得 x = 4。
or log base 10. For x in the exponent, bring it down using the power law. Example: solve 5ˣ = 13. Taking ln on both sides: ln(5ˣ) = ln 13 → x ln 5 = ln 13 → x = ln 13 / ln 5. Alternatively, use log base 10.
如果底数不易统一，则对两边取对数，通常取自然对数或以 10 为底的对数。对指数上的未知数，利用幂法则将其“拉下来”。例如解 5ˣ = 13，两边取 ln：ln(5ˣ) = ln 13 → x ln 5 = ln 13 → x = ln 13 / ln 5。也可用 log₁₀ 求解。
Mixed Exercise 14 often mixes these approaches, including disguised quadratics like 3²ˣ + 3ˣ − 2 = 0. Substituting y = 3ˣ on: extraneous solutions can arise, especially when log arguments become negative or zero.
混合练习 14 常会将这两种方法融合考察，包括伪装成二次方程的类型，如 3²ˣ + 3ˣ − 2 = 0。令 y = 3ˣ 代换后转化为二次方程。解出后一定要代入原方程验证真伪，特别是当对数真数出现负数或零时，会产生增根。
5. Solving Logarithmic Equations | 解对数方程
using the laws. Then rewrite in exponential form to eliminate the log. Alternatively, if the equation equates two logs with the same base, you can equate their arguments: logₐ M = logₐ N ⇒ M = N, provided M, N > 0.
对数方程含有一个或多个对数项。首先通常将所有对数项集中到一边，运用对数法则压缩成单个对数。然后改用指数形式以消去对数。另一种情况，如果等式两边是同底的两个对数，则可直接令真数相等：logₐ M = logₐ N ⇒ M = N，前提是 M, N > 0。
.70 or x ≈ −2.70. The negative solution must be rejected because it makes (x−2) negative, and logs of negative numbers are undefined. So only x ≈ 3.70 is valid. Always state the domain restrictions before solving.
以方程 log₂ (x + 1) + log₂ (x − 2) = 3 为例。用加法法则得 log₂ [(x+1)(x−2)] = 3。改写为 2³ = (x+1)(x−2) → 8 = x² − x − 2 → x² − x − 10 = 0，解得 x ≈ 3.70 或 x ≈ −2.70。负根必须舍去，因为它会使 (x−2) 为负，而对数的真数不能为负。故仅有 x ≈ 3.70 成立。解方程前一定要先明确定义域限制。
ck extraneous solutions. Another common type is logₐ (something) = k, which becomes aᵏ = something. Remember to keep track of the base throughout.
在混合练习 14 中，你还可能遇到两边都带有对数项且系数不同的方程。这时需做代数整理并检验增根。另一种常见类型是 logₐ (某式) = k，转化为 aᵏ = 某式。过程中要时刻注意底数。
6. Exponential Growth and Decay Models | 指数增长与衰减模型
P = P₀ eᵏᵗ, where P₀ is the initial quantity, k is the growth (positive) or decay (negative) constant, and t is time. The base e is used because continuous growth can be described elegantly with it.
指数函数可以对许多现实情形建模，如人口增长、放射性衰变和复利。一般形式为 P = P₀ eᵏᵗ，其中 P₀ 是初始值，k 是增长（正）或衰减（负）常数，t 是时间。底数 e 的使用使得连续增长可以被优雅地描述。
ng natural logs. For half-life problems, set P = ½ P₀ and solve for t. The relationship between half-life t½ and decay constant k is ln(½) = −k t½, leading to k = ln 2 / t½.
你可能会被要求根据两个数据点求 k，或预测未来值。将已知数值代入公式，然后用自然对数解出未知数。对于半衰期问题，令 P = ½ P₀ 并解出 t。半衰期 t½ 与衰减常数 k 的关系是 ln(½) = −k t½，从而得出 k = ln 2 / t½。
s consistent and to give answers to an appropriate degree of accuracy. A typical question: ' reases from 500 to 1200 in 6 hours. Find the continuous growth rate and the doubling time.' Solve 1200 = 500 e6k to get k, then doubling time = (ln 2)/k.
混合练习 14 常包含一个实际背景问题，需要你对常数加以解释。注意保持单位一致，并给出适当精度的答案。一道典型题目是：“一个细菌群落在 6 小时内从 500 增长到 1200。求连续增长率和倍增时间。”通过 1200 = 500 e6k 解出 k，然后倍增时间 = (ln 2)/k。
7. Logarithmic Scales and Data Linearisation | 对数标度与数据线性化
When experimental data follows an exponential relationship y = a bˣ or a power law y = a xⁿ, logarithms can transform them into linear form. For y = a bˣ ives log y = log a + n log x, so plotting log y against log x gives a straight line with gradient n and intercept log a.
当实验数据遵循指数关系 y = a bˣ 或幂律 y = a xⁿ 时，利用对数可以将它们转化为线性形式。对于 y = a bˣ，两边取对数得 log y = log a + x log b。将 log y 对 x 作图会得到一条直线，斜率为 log b，截距为 log a。对于 y = a xⁿ，取对数得 log y = log a + n log x，因此将 log y 对 log x 作图会得到一条斜率为 n、截距为 log a 的直线。
This technique is known as data linearisation and is a key exam skill. You may be given a table of x and y values and asked to complete a table of log y, then plot and find the unknown constants a and b (or n). Use the gradient and intercept from the straight-line graph to determine the constants, and then rewrite the original relationship.
这一技巧称为数据线性化，是考试中的一项关键技能。你可能会拿到一张 x 与 y 的数据表，被要求完成 log y 表格，然后绘图并求出未知常数 a 和 b（或 n）。利用直线图的斜率和截距来确定常数，再还原出原始关系。
Mixed Exercise 14 typically contains at least one question on this topic. It tests your ability to use the laws of logs in a practical context. Be careful with axis labels: 'log y' on the vertical axis means you plot log₁₀ y or ln y depending on the instructions. If the question says 'plot log y' without specifying the base, log₁₀ is conventionally assumed.
混合练习 14 通常至少有一道与此相关的题目。它考查你在实际情境中运用对数法则的能力。注意坐标轴标签：纵轴上的 'log y' 指根据题目要求绘制 log₁₀ y 或 ln y。如果题目只说 'plot log y' 而未指定底数，通常默认以 10 为底。
8. Equations Involving eˣ and ln | 涉及 eˣ 与 ln 的方程
ral logarithm, ln, is the inverse of the exponential function with base e. This means ln(eˣ) = x for all real x, and eln x se cancellation properties are extremely useful when solving equations that involve both eˣ and ln. When you see an equation like e²ˣ⁺¹ = 5, taking the natural log of both sides gives 2x + 1 = ln 5 directly.
自然对数 ln 是以 e 为底的指数函数的逆运算。这意味着对所有实数 x，有 ln(eˣ) = x；对 x > 0，有 eln x = x。这些相消性质在解包含 eˣ 和 ln 的方程时极为有用。当你看到类似 e²ˣ⁺¹ = 5 的方程，两边取自然对数可直接得到 2x + 1 = ln 5。
he variable from inside a log. Remember to check that the argument 3x − 4 remains positive; if the solution yields 3x − 4 ≤ 0, reject it.
反过来，如果遇到 ln(3x − 4) = 2，两边以 e 为底取指数得 3x − 4 = e²，然后解出 x。这是把变量从对数中“解放”出来的标准方法。记住检验真数 3x − 4 是否为正；如果解得 3x − 4 ≤ 0，应舍去。
Advanced problems combine multiple exponentials and logs. For example, solve eˣ + e⁻ˣ = 3. Multiply through by eˣ to get e²ˣ + 1 = 3eˣ, which is a quadratic in eˣ: (eˣ)² − 3eˣ + 1 = 0. Solve using the quadratic formula: eˣ = (3 ± √5)/2. Since eˣ > 0, both roots are acceptable, and x = ln((3 ± √5)/2). Mixed Exercise 14 will stretch you with such multi-step manipulations.
较难的问题会将多个指数和对数组合起来。例如解 eˣ + e⁻ˣ = 3。两边同乘 eˣ 得到 e²ˣ + 1 = 3eˣ，这是关于 eˣ 的二次方程：(eˣ)² − 3eˣ + 1 = 0。用求根公式得 eˣ = (3 ± √5)/2。由于 eˣ > 0，两根均可取，因此 x = ln((3 ± √5)/2)。混合练习 14 会通过这类多步操作挑战你的能力。
9. Graphs of Logarithmic Functions | 对数函数的图像
e is the y-axis (x = 0). For 0 < a < 1, the graph decreases. The natural log graph y = ln x is the inverse of y = eˣ . Understanding these graphs helps to visualise the domain and behaviour of log functions.
函数 y = logₐ x (a > 1) 的图像经过 (1, 0)，完全位于 y 轴右侧（定义域 x > 0），并缓慢递增。其垂直渐近线是 y 轴 (x = 0)。若 0 < a < 1，图像递减。自然对数图像 y = ln x 是 y = eˣ 的反函数，关于直线 y = x 对称。理解这些图像有助于直观地把握对数函数的定义域和变化趋势。
asymptote x = 3. y = −ln x reflects in the x-axis. y = ln(2x) compresses horizontally. You may be asked to sketch such transformations or to find intercepts with the axes.
函数图像变换对对数函数同样适用。y = log₂ (x − 3) 将基本图像向右平移 3 个单位，定义域变为 x > 3，渐近线为 x = 3。y = −ln x 关于 x 轴反射。y = ln(2x) 沿水平方向压缩。你可能需要画出此类变换的图像，或求出与坐标轴的交点。
In Mixed Exercise 14, graphical questions often involve finding the intersection of an exponential and a logarithmic function, or using the graph to solve inequalities like log₂ (x + 1) > 3. Sketch both curves and look for the regions. Always label key points and asymptotes clearly.
在混合练习 14 中，图像题常涉及求指数函数与对数函数的交点，或利用图像解不等式，如 log₂ (x + 1) > 3。画出两条曲线并观察区域。要清晰地标注关键点和渐近线。
10. Common Pitfalls and Revision Tips | 常见错误与复习建议
Many students lose marks by forgetting that log (x + y) is not log x + log y, or by incorrectly applying the power law inside a log of a sum. Always pause and check whether an expression can be split. Another typical mistake is ignoring domain restrictions when solving log equations. After finding algebraic solutions, always test them in the original equation or check that the arguments of any logs remain positive.
许多学生因误以为 log (x + y) 等于 log x + log y，或在和的求对中误用幂法则而失分。遇到此类表达式务必停顿检查是否可以拆分。另一个典型错误是解对数方程时忽略定义域限制。求出代数解后，一定要代入原方程检验，或确认所有对数的真数保持为正。
When dealing with eˣ and ln, do not confuse their properties. While eln x = x, the expression ln(eˣ + eʸ r argument is exactly the inverse function. Also, be precise with calculator work: use the 'ln' button for natural logs and know when to store exact values to avoid rounding errors.
处理 eˣ 和 ln 时，不要混淆它们的性质。虽然 eln x = x，但 ln(eˣ + eʸ) 无法化简。仅当整个指数或真数恰好为对方的逆函数时，才能使用相消关系。同时，计算器使用要精准：用 'ln' 键处理自然对数，并知道何时保留精确值以避免舍入误差。
To revise effectively for Mixed Exercise 14, practise a wide range of problems from previous exam papers and textbook exercises. Ensure you can explain the reasoning behind each step, not just memorise procedures. Create a summary card with the three log laws, the change-of-base formula, and the cancellation properties. Use it to test yourself until the recall is instant. Consistent practice with checking domains will greatly improve your accuracy.
要高效复习混合练习 14，应广泛练习往年真题和课本习题。确保能解释每一步的理由，而不仅仅是记住步骤。制作一张摘要卡，写上三条对数运算法则、换底公式以及相消性质。用它自我测试，直到能瞬间回忆。坚持练习并检查定义域将极大提升你的准确性。
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