Solving Equations Using Logarithms | 使用对数解方程

📚 Solving Equations Using Logarithms | 使用对数解方程

Logarithms are the natural inverse of exponential functions, making them indispensable when the unknown appears in an exponent. In A-Level Mathematics, you will meet a wide range of equations – from simple exponential forms like 2^x ations and hidden quadratics. This article builds your skills step by step, covering all the major techniques required by the Edexcel specification, with careful attention to algebraic manipulation, domain restrictions, and checking for extraneous roots.

对数是指数函数的自然逆运算，因此当未知数出现在指数位置时，对数便不可或缺。在 A-Level 数学中，你会遇到各种方程——从 2^x = 7 这样简单的指数形式，到更加复杂的对数方程和隐藏二次型。本文逐层构建你的解题能力，覆盖 Edexcel 考纲要求的所有主要技巧，并特别关注代数变形、定义域限制与增根的检验。

1. Why Logarithms Are Essential for Solving Equations | 为什么解方程需要对数？

If an equation contains a term like a^x (with a > 0, a ≠ 1), we cannot isolate x using just the four basic operations. The logarithm enables us to “bring the power down” via the identity log(a^p tial equation into a linear or quadratic equation that can be solved using familiar methods.

如果方程中含有 a^x 这样的项（a > 0，a ≠ 1），我们无法仅用四则运算把 x 分离出来。对数通过恒等式 log(a^p) = p log a 将幂“拉下来”，从而把指数方程转化为可以用熟悉方法求解的线性或二次方程。

Commonly used logarithms are log₁₀ (common log) and log_e or ln (natural logarithm). The natural logarithm is especially useful because the derivative of e^x is e^x, which appears frequently in growth and decay problems.

常用的对数有 log₁₀（常用对数）和 logₑ（即 ln，自然对数）。自然对数尤其有用，因为 e^x 的导数还是 e^x，这一函数在增长与衰减问题中频繁出现。

---

2. Solving Simple Exponential Equations a^x = b | 解简单的指数方程 a^x = b

To solve a^x = b, take logarithms of both sides. Any base can be chosen, but ln or log₁₀ is typical. Using the power rule:

要解 a^x = b，可以对两边取对数。底数可任意选择，但通常用 ln 或 log₁₀。利用幂的法则：

ln(a^x) = ln b ⇒ x ln a = ln b ⇒ x = ln b / ln a

Notice that the answer is simply log_ab expressed in natural logs. For example, solve 3^x = 20.

注意到答案其实就是用自然对数表示的 log_ab。例如，解方程 3^x = 20。

x = ln 20 / ln 3 ≈ 2.7268 (4 d.p.)

Always keep the answer exact unless a decimal approximation is requested. The exact form is ln 20 / ln 3.

除非题目要求近似值，否则保持精确形式。精确形式为 ln 20 / ln 3。

---

3. Using Natural Logarithms (ln) in Growth and Decay | 使用自然对数处理增长与衰减

tions that model continuous growth or decay are often written with base e, such as P = P₀e^kt. Solving for t when P is given involves isolating e^kt and then taking ln of both sides.

描述连续增长或衰减的方程通常以 e 为底，如 P = P₀e^kt。当 P 已知求 t 时，需先分离出 e^kt，再对两边取 ln。

For instance, if 500 = 200e^0.03t, divide by 200 to obtain 2.5 = e^0.03t, then:

例如，500 = 200e^0.03t，除以 200 得 2.5 = e^0.03t，然后：

ln 2.5 = 0.03t ⇒ t = ln 2.5 / 0.03

This exact value is the answer, and can be evaluated to approximately 30.54. Note that ln and e are inverses, so ln(e^k) = k.

精确值即为答案，计算后约为 30.54。注意 ln 与 e 互逆，因此 ln(e^k) = k。

---

4. Equations Requiring a Change of Base | 需要换底公式的方程

Occasionally you might see an equation like log₄(x) + log₁₆ to handle if you change all logarithms to the same base using the change-of-base formula.

有时你会遇到 log₄(x) + log₁₆(x) = 3 这样的方程。如果使用换底公式将所有对数化成同底，处理起来会更简便。

log_ab = log_cb / log_ca

becomes (1/2 + 1/4) log₂ x = 3 ⇒ (3/4) log₂ x = 3 ⇒ log₂ x = 4, so x = 2⁴ = 16.

选择底数 2 很方便，因为 4 = 2²，16 = 2⁴。于是 log₄(x) = log₂(x)/log₂(4) = (log₂ x)/2，log₁₆(x) = (log₂ x)/4。方程化为 (1/2 + 1/4)log₂ x = 3，得 (3/4)log₂ x = 3，log₂ x = 4，从而 x = 2⁴ = 16。

---

5. Solving Equations with Exponential Terms on Both Sides | 解两边都有指数项的方程

When an equation contains different bases raised to expressions in x, e.g. 3^2x+1 = 5^x−3, taking logarithms (often ln) of both sides linearises the equation.

当方程两边不同底数的指数中含有 x 时，例如 3^2x+1 = 5^x−3，对两边取对数（通常取 ln）可将方程线性化。

(2x+1) ln 3 = (x−3) ln 5

= −(3 ln 5 + ln 3)/(2 ln 3 − ln 5). This exact expression can be simplified using laws of logs, but is acceptable as the final answer.

展开并收集含 x 的项：2x ln 3 − x ln 5 = −3 ln 5 − ln 3。左边提取 x：x(2 ln 3 − ln 5) = −(3 ln 5 + ln 3)。于是 x = −(3 ln 5 + ln 3)/(2 ln 3 − ln 5)。该精确表达式可借助对数法则化简，但直接保留此形式也可作为最终答案。

Be careful with signs during collection; double-check by substituting the approximate value back into the original equation.

移项时注意正负号；可将近似值代回原方程进行检验。

---

6. Solving Basic Logarithmic Equations | 解基本对数方程

An equation such as log₃(x) = 4 can be solved by converting to exponential form: x = 3⁴ = 81. More generally, log_a(f(x)) = b ⇒ f(x) = a^b.

对于 log₃(x) = 4 这样的方程，可转化为指数形式：x = 3⁴ = 81。更一般地，logₐ(f(x)) = b ⇒ f(x) = a^b。

If the equation contains more than one logarithm on one side, like log₂(x+1) = 3, isolate the logarithm first if necessary, then rewrite in exponential form.

如果方程一边含有不止一个对数，如 log₂(x+1) = 3，必要时先将对数分离出来，再改写为指数形式。

Remember that the argument of a logarithm must be strictly positive. After finding x, check that it satisfies f(x) > 0. For log_a(x), we also require a > 0, a ≠ 1.

切记对数的真数必须为正。求出 x 后，需验证其满足 f(x) > 0。对于 logₐ(x)，还要求 a > 0 且 a ≠ 1。

---

广告 / Sponsored

7. Applying Logarithm Laws to Simplify Equations | 应用对数律简化方程

ing the laws: log A + log B = log(AB), log A − log B = log(A/B), and n log A = log(Aⁿ).

对数方程常需借助对数律进行合并或展开：log A + log B = log(AB)，log A − log B = log(A/B)，以及 n log A = log(Aⁿ)。

For example, solve log₂(x) + log₂(x−2) = 3. Combine the left side: log₂(x(x−2)) = 3. Then rewrite: x(x−2) = 2³ nus;2 > 0, so x > 2. Thus x = 4 is the only valid solution; x = −2 must be rejected.

例如，解 log₂(x) + log₂(x−2) = 3。合并左边：log₂(x(x−2)) = 3。改写为指数形式：x(x−2) = 2³ = 8。化简得 x² − 2x − 8 = 0，解得 x = 4 或 x = −2。检验定义域：x > 0 且 x−2 > 0，因此 x > 2。故 x = 4 是唯一有效解，x = −2 应舍去。

This demonstrates why domain checking is essential for logarithmic equations.

这说明了为什么解对数方程时检查定义域至关重要。

---

8. Quadratic Exponential Equations (Hidden Quadratics) | 二次指数方程（隐藏二次型）

Equations of the form p a^2x + q a^x a constant a > 0, are called “hidden quadratics”. Make the substitution y = a^x (where y > 0). The equation becomes p y² + q y + r = 0, a standard quadratic.

形如 p a²ˣ + q aˣ + r = 0（a > 0 且为常数）的方程称为“隐藏二次型”。作代换 y = aˣ（其中 y > 0），方程化为 p y² + q y + r = 0，即标准二次方程。

For instance, solve e^2x − 5e^x + 6 = 0. Let y = e^x y−3) = 0, so y = 2 or y = 3. Both are positive, hence acceptable. Back-substitute: e^x = 2 ⇒ x = ln 2, and e^x = 3 ⇒ x = ln 3.

例如，解 e²ˣ − 5eˣ + 6 = 0。令 y = eˣ（y > 0），则有 y² − 5y + 6 = 0，得 (y−2)(y−3) = 0，故 y = 2 或 y = 3。两者皆为正，均可接受。回代：eˣ = 2 ⇒ x = ln 2，eˣ = 3 ⇒ x = ln 3。

This technique works for any constant base, such as 2^2x + 3·2^x − 4 = 0. Remember to reject any negative solutions for y because a^x > 0 for all real x.

此方法适用于任意常底数，如 2²ˣ + 3·2ˣ − 4 = 0。记住舍弃任何负的 y 解，因为对所有实数 x 均有 aˣ > 0。

---

9. Equations Involving Both Logarithms and Exponentials | 同时包含对数和指数的方程

Some equations mix logarithmic and exponential parts, e.g. x e^x = 5 or ln(x+1) = e^x. For x e^x ; numerical methods or the Lambert W function would be needed, but for A-Level purposes, you may be asked to show that a solution lies in a given interval, or to solve using iterative methods. However, an equation like e^{ln x} = 7 simplifies immediately because e^{ln x} = x for x > 0, giving x = 7.

有些方程同时含有对数和指数，如 x eˣ = 5 或 ln(x+1) = eˣ。对于 x eˣ = 5，常规代数方法无效，需借助数值方法或朗伯 W 函数，但在 A-Level 范围内，可能会要求证明解在某个区间内，或使用迭代法求解。然而，像 e^{ln x} = 7 这样的方程可立刻化简，因为对于 x > 0 有 e^{ln x} = x，从而得 x = 7。

Another common type is log_a(k e^bx) = c, which simplifies to log_a k + bx log_a e = c using log properties. Solve for x directly.

另一种常见类型是 logₐ(k e^bx) = c，利用对数性质可化为 logₐ k + bx logₐ e = c，直接解出 x。

Always exploit inverse relations: ln(e^x) = x and e^{ln x} = x (for x > 0). These can instantly simplify many expressions.

务必善用逆关系：ln(eˣ) = x 以及 e^{ln x} = x（x > 0）。它们可以瞬间化简许多表达式。

---

10. Checking for Extraneous Solutions and Domain Restrictions | 检验增根与定义域限制

Logarithmic equations are particularly prone to extraneous solutions because the transformations used (combining logs, squaring) can introduce values that do not satisfy the original domain. Three domain conditions must be checked for any log_a(f(x)):

对数方程特别容易产生增根，因为所用变换（合并对数、平方）可能引入不满足原定义域的值。对于任何 logₐ(f(x))，必须检查以下三个定义域条件：

• Base: a > 0 and a ≠ 1 (底数：a > 0 且 a ≠ 1)
• Argument: f(x) > 0 (真数：f(x) > 0)
• If the original equation contains multiple logs, each argument must be positive. (若原方程含有多个对数，每个真数都必须为正。)

After solving, substitute each candidate back into the original equation. If any logarithm becomes undefined or the equation does not hold, discard that root. Do not skip this step; Edexcel mark schemes often award marks for rejecting an invalid solution.

求解后，将每个候选根代回原方程。若任一对数变为无定义或等式不成立，则舍弃该根。切勿忽略此步骤；Edexcel 评分标准常对舍去无效解给予相应分数。

---

11. Graphical Insights and Connections | 图像洞察与联系

Visualising equations can deepen understanding. The equation a^x be seen as the intersection of the horizontal line y = b with the exponential curve y = a^x. Taking logs is equivalent to finding the x-coordinate using the inverse function log_a.

可视化方程可加深理解。方程 aˣ = b 可视为水平线 y = b 与指数曲线 y = aˣ 的交点。取对数等同于使用反函数 logₐ 求交点的 x 坐标。

l methods (or an iteration based on a rearrangement) may be employed to locate the root because it cannot be solved algebraically. Being comfortable with the shapes of exponential and logarithmic graphs aids in reasoning about the number of solutions and their approximate locations.

对于 ln(x−1) = 2x−5 这样的对数方程，由于无法代数求解，可能会用到图像法（或基于重排的迭代法）定位根。熟悉指数与对数图像的形状，有助于判断解的个数及其大致位置。

Understanding that y = ln x is the reflection of y = e^x in the line y = x also reinforces why ln(e^x) = x and e^{ln x} = x.

理解 y = ln x 是 y = eˣ 关于直线 y = x 的反射，也能强化为什么 ln(eˣ) = x 以及 e^{ln x} = x。

---

12. Key Takeaways and Exam Tips | 关键要点与考试技巧

Mastering equations with logarithms boils down to a few golden rules:

掌握含对数的方程可归结为以下几条黄金法则：

• To remove an exponential, take logs (usually ln). (去除指数，取对数——通常为 ln。)
• To remove a logarithm, rewrite in exponential form. (去除对数，改写为指数形式。)
• Use log laws only when it simplifies the equation; always be aware of domain changes. (仅在能简化方程时使用对数律；始终注意定义域的变化。)
• In hidden quadratics, substitute y = a^x and remember y > 0. (在隐藏二次型中，设 y = aˣ 并谨记 y > 0。)
• Always verify your solutions by checking the original domain, and explicitly reject invalid roots. (始终通过检查原定义域验证解，并明确舍去无效根。)

In the exam, show clear steps: write down the log operation you are applying, use exact forms unless told otherwise, and comment on the validity of each solution. These habits will earn you full method marks and avoid careless errors.

在考试中，步骤要清晰：写下你所取的对数运算，除非另有说明否则使用精确形式，并对每个解的有效性加以说明。这些习惯将助你拿到完整的步骤分，避免粗心错误。

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）