A-Level Maths: Logarithmic Transformation of Non-Linear Data | A-Level 数学：对数变换处理非线性数据

📚 A-Level Maths: Logarithmic Transformation of Non-Linear Data | A-Level 数学：对数变换处理非线性数据

In A-Level mathematics, many real-world data sets do not follow a straight line. Instead, they may follow exponential or power-law patterns. To analyse such data, we use logarithmic transformations to turn curved relationships into straight lines. This makes it easier to estimate parameters and make predictions.

在 A-Level 数学中，许多真实数据集并不服从直线关系，而可能服从指数或幂律模式。为了分析这类数据，我们使用对数变换把曲线关系转化为直线。这样更容易估计参数并进行预测。

1. Why Logarithms? | 为什么使用对数？

Logarithms have three key properties that make them ideal for linearising data. First, log(ab) = log a + log b. Second, log(aⁿ) = n log a. Third, logₐ(a) = 1 and logₐ(1) = 0. These properties allow us to separate variables and convert multiplication and exponentiation into addition and multiplication.

对数有三个关键性质，使其非常适合线性化数据。第一，log(ab) = log a + log b。第二，log(aⁿ) = n log a。第三，logₐ(a) = 1，logₐ(1) = 0。这些性质使我们可以分离变量，并将乘法和幂运算转化为加法和乘法。

log(ab) = log a + log b, log(aⁿ) = n log a

2. Exponential Model y = kbˣ | 指数模型 y = kbˣ

Consider data modelled by y = k bˣ, where k and b are constants. Taking the natural logarithm of both sides gives ln y = ln k + x ln b. This has the form Y = mX + c, where Y = ln y, X = x, m = ln b, and c = ln k. Therefore, a plot of ln y against x will produce a straight line.

考虑由 y = k bˣ 建模的数据，其中 k 和 b 为常数。对两边取自然对数得到 ln y = ln k + x ln b。这是 Y = mX + c 的形式，其中 Y = ln y，X = x，m = ln b，c = ln k。因此，绘制 ln y 对 x 的图将得到一条直线。

ln y = ln k + x ln b

The slope of the line is ln b, and the y-intercept is ln k. If the natural logarithm is used, recovery of b and k requires exponentiating: b = e^slope and k = e^intercept.

直线的斜率是 ln b，y

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