📚 Logarithms for Linearising Non-Linear Data | 用对数处理非线性数据的技巧
In A-Level mathematics, real-world data rarely follows a perfect straight line. Yet the most convenient way to analyse a relationship between two variables is still the linear form y = mx + c: a straight line has a constant gradient, a single intercept, and is easy to extend and interpret. Logarithms allow us to convert two of the most common non-linear models — the exponential model y = abˣ and the power model y = axⁿ — directly into straight lines. This guide explains the technique, works through full exam-style examples, and highlights the traps that examiners love to set.
在 A-Level 数学中，真实世界的数据很少恰好落在一条完美的直线上。然而，分析两个变量之间关系最方便的方式仍然是直线形式 y = mx + c：直线具有恒定的斜率、单一的截距，并且容易延伸和解读。对数让我们能够把两类最常见的非线性模型——指数模型 y = abˣ 和幂模型 y = axⁿ——直接变换为直线。本指南将讲解这一技巧，演示完整的考试风格例题，并指出考官最爱设置的陷阱。
1. Why Transform the Data? | 一、为什么要对数据做变换
A curved scatter graph is difficult to describe precisely. Its slope changes at every point, so a single numerical value for the gradient cannot be reported. Fitting a smooth curve by eye is unreliable, and comparing two different curved relationships is almost impossible. A straight line, by contrast, is fully described by just two numbers: the gradient m and the intercept c.
曲线散点图难以精确描述。它的斜率在每一点都在变化，因此无法报告单一的梯度数值。凭肉眼拟合光滑曲线并不可靠，比较两条不同的曲线关系也几乎不可能。相比之下，一条直线只需两个数字即可完全描述：斜率 m 和截距 c。
Taking logarithms turns two important families of curves into straight lines:
对数的引入可以把两类重要的曲线族转化为直线：
- Exponential relationships y = abˣ become linear when ln y is plotted against x.
- 指数关系 y = abˣ 在绘制 ln y 对 x 的图时变为直线。
- Power relationships y = axⁿ become linear when log y is plotted against log x.
- 幂关系 y = axⁿ 在绘制 log y 对 log x 的图时变为直线。
Once the graph is a straight line, you can find m and c from the plot or from least-squares calculations, then convert them back to the original parameters a and b (or a and n).
一旦图像成为直线，就可以从图或最小二乘计算中求出 m 和 c，再把它们转换回原始参数 a、b（或 a、n）。
2. The Exponential Model y = abˣ | 二、指数模型 y = abˣ
Consider the exponential model y = abˣ, where a and b are positive constants. Taking the natural logarithm of both sides gives:
考虑指数模型 y = abˣ，其中 a、b 为正数。对两边取自然对数得到：
ln y = ln a + x ln b
This has exactly the same structure as y = mx + c. If we treat ln y as the vertical coordinate and x as the horizontal coordinate, the graph is a straight line with:
这与 y = mx + c 的结构完全相同。若把 ln y 当作纵坐标，把 x 当作横坐标，图像是一条直线，且：
- Gradient m = ln b, so b = eᵐ.
- 斜率 m = ln b，因此 b = eᵐ。
- Intercept c = ln a, so a = eᶜ.
- 截距 c = ln a，因此 a = eᶜ。
The exponent x is not transformed; only the dependent variable y is logged. This is why the technique is sometimes called a "log-linear" plot.
指数 x 不变换，只有因变量 y 被取对数。这就是为什么该技巧有时被称为"对数-线性"图。
3. The Power Model y = axⁿ | 三、幂模型 y = axⁿ
For the power model y = axⁿ, taking logarithms of both sides — using any consistent base, commonly base 10 — gives:
对于幂模型 y = axⁿ，对两边取对数（使用一致的底数，通常取 10）得到：
log y = log a + n log x
Now both variables must be transformed. Plotting log y against log x produces a straight line because log y is a linear function of log x:
此时两个变量都需要变换。绘制 log y 对 log x 的图像会得到一条直线，因为 log y 是 log x 的线性函数：
- Gradient m = n, the power itself.
- 斜率 m = n，即幂指数本身。
- Intercept c = log a, so a = 10ᶜ.
- 截距 c = log a，因此 a = 10ᶜ。
A log-log plot has a beautiful property: the gradient is the power n. If n = 2 you see a parabola as a straight line of slope 2; if n = ½ the square-root curve becomes a straight line of slope 0.5.
双对数图有一个极佳的性质：斜率就是幂指数 n。如果 n = 2，抛物线变成斜率为 2 的直线；如果 n = ½，平方根曲线变成斜率为 0.5 的直线。
4. Recovering the Parameters | 四、由斜率与截距反推出参数
The table below summarises both transformations and the conversions you must perform. A common exam mistake is to quote m or c directly as the original parameter without applying the inverse logarithm.
下表总结了两类变换及必须执行的换算。一个常见考试错误是直接把 m 或 c 当作原始参数，而没有取对数还原。
| Model | Plot | Gradient | Intercept | Parameters |
| y = abˣ | ln y vs x | m = ln b | c = ln a | b = eᵐ, a = eᶜ |
| y = axⁿ | log y vs log x | m = n | c = log a | n = m, a = 10ᶜ |
In both cases, if natural logarithms are used instead of base-10 logarithms, replace 10ᶜ with eᶜ. The gradient-to-parameter conversion changes accordingly: for y = abˣ with ln, b = eᵐ; for y = axⁿ with ln, a = eᶜ.
在两种情况下，如果使用自然对数而不是以 10 为底的对数，把 10ᶜ 换成 eᶜ 即可。斜率到参数的换算也随之改变：对 y = abˣ 取 ln 时 b = eᵐ；对 y = axⁿ 取 ln 时 a = eᶜ。
5. ln or log₁₀? Which Base Should I Use? | 五、ln 还是 log₁₀？应该用哪个底数？
Mathematically, any base works as long as you are consistent. In practice, the choice follows the structure of the model. For exponential models y = abˣ, natural logarithms are natural because the inverse of the exponential function eˣ is ln. For power models y = axⁿ, base-10 logarithms are common in past exam papers because they avoid writing the base every time.
从数学上说，任何底数都可以，只要保持一致。实践中，选择取决于模型的结构。对于指数模型 y = abˣ，自然对数更自然，因为指数函数 eˣ 的反函数正是 ln。对于幂模型 y = axⁿ，历年真题中常用以 10 为底的对数，因为可以省去每次写出底数。
There is one hard rule: never mix bases in the same calculation. If you plot ln y on the vertical axis, the gradient must be read as ln b, not log b. If you plot log₁₀ y, the intercept gives log₁₀ a, and a = 10ᶜ.
有一条硬性规则：绝不在同一计算中混用底数。如果纵轴是 ln y，斜率就必须解读为 ln b，而不是 log b。如果绘制 log₁₀ y，截距给出 log₁₀ a，则 a = 10ᶜ。
6. Worked Example 1: Exponential Growth | 六、例题 1：指数增长
The population P of a bacterial culture is recorded every hour. The results are shown below. It is believed that P = abᵗ.
某种细菌培养物的数量 P 每小时记录一次，结果如下表。已知 P = abᵗ。
| t (hours) | 0 | 1 | 2 | 3 | 4 |
| P | 200 | 420 | 880 | 1850 | 3890 |
| ln P | 5.30 | 6.04 | 6.78 | 7.52 | 8.27 |
Since P = abᵗ, we have ln P = ln a + t ln b. Plotting ln P against t gives a straight line. Using the first and last points:
因为 P = abᵗ，所以 ln P = ln a + t ln b。绘制 ln P 对 t 的图像得到一条直线。取首末两点：
gradient = (8.27 − 5.30) / (4 − 0) = 0.7425
Therefore ln b ≈ 0.743, so b = e⁰·⁷⁴³ ≈ 2.10. The intercept is ln P at t = 0, which is 5.30, so a = e⁵·³⁰ ≈ 200.
因此 ln b ≈ 0.743，所以 b = e⁰·⁷⁴³ ≈ 2.10。截距是 t = 0 时的 ln P，即 5.30，故 a = e⁵·³⁰ ≈ 200。
P = 200 × 2.10ᵗ
Notice that 2.10 is the growth factor per hour: each hour the population multiplies by about 2.1. Check: at t = 2, P = 200 × 2.10² = 882, which agrees with the observed 880.
注意 2.10 是每小时的增长率：每个小时数量乘以约 2.1。检验：t = 2 时，P = 200 × 2.10² = 882，与观测值 880 吻合。
7. Worked Example 2: A Power Law | 七、例题 2：幂律关系
The period T of a simple pendulum is measured for different lengths L. The data is believed to follow T = kLⁿ. Find the values of n and k.
测量不同摆长 L 下单摆的周期 T，数据被认为符合 T = kLⁿ。求 n 和 k 的值。
| L (m) | 0.2 | 0.4 | 0.6 | 0.8 | 1.0 |
| T (s) | 0.90 | 1.27 | 1.55 | 1.79 | 2.01 |
| log₁₀ L | −0.699 | −0.398 | −0.222 | −0.097 | 0 |
| log₁₀ T | −0.0458 | 广告 / Sponsored 培生爱德思考试教材 统计数学3 Edexcel International A Level Mathematics S… 数学 备考教辅 · 正版现货 · 与本文内容相关 查看详情 / View on Taobao参考价 ¥156.0
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