指数与对数：增长和尺度Exponentials & Logarithms: Growth and Scale
怎样把成倍变化转换成容易比较的加法尺度？How can multiplicative change become an easier-to-compare additive scale?
前置知识：Prerequisites: B01：函数、坐标与参数。B01: functions, coordinates and parameters.
复习前置：Review prerequisites: B01 · 函数、坐标与参数Functions, Coordinates & Parameters
学完后你能：After this lesson you can: 辨认成倍增长，读懂对数坐标，并判断对数何时有定义。Recognize multiplicative growth, read a logarithmic axis, and identify when a logarithm is defined.
动手观察Explore
切换纵轴，观察同一组数据；把检验值设为 0，查看对数的定义域。Switch the vertical axis for the same data; set the probe value to 0 to inspect the logarithm domain.
从观察到解释From observation to explanation
先看增长First, observe growth
1、2、4、8 每一步乘 2；1、2、3、4 每一步加 1。前者是成倍增长，后者是等差增长。用普通纵轴画前者，后面的距离越来越大。In 1, 2, 4, 8 each step multiplies by 2; in 1, 2, 3, 4 each step adds 1. The first grows multiplicatively; the second grows by equal increments. On an ordinary axis the former spreads out.
换一把尺Change the ruler
对数回答“底数要乘多少次，才能得到这个数？”例如 2³ = 8，所以 log₂ 8 = 3。对数纵轴上 1 → 2 → 4 → 8 是相等间隔，因为每次都乘 2。数据本身没有变。A logarithm asks “what power of the base produces this number?” For example, 2³ = 8, so log₂ 8 = 3. On a logarithmic vertical axis 1 → 2 → 4 → 8 are equally spaced because each step multiplies by 2. The data have not changed.
乘法变成加法Multiplication becomes addition
正数 u、v 满足 log_c(uv) = log_c u + log_c v。例如 log₂(4 × 8) = 2 + 3 = 5。负数和零不能取实数对数，不能画在实数对数轴上。For positive u and v, log_c(uv) = log_c u + log_c v. For example log₂(4 × 8) = 2 + 3 = 5. Zero and negative numbers have no real logarithm and cannot be put on a real log axis.
公式与符号Formula and symbols
c 是底数（c > 0 且 c ≠ 1），x 是指数，y 是结果；实数对数要求 y > 0。ln 使用底数 e ≈ 2.718，log₂ 使用底数 2。c is the base (c > 0 and c ≠ 1), x the exponent, and y the result; real logarithms require y > 0. ln has base e ≈ 2.718, while log₂ has base 2.
留意这个区别Watch for this
坐标轴等距表示等倍，不是等差。比较信息量或损失时，必须说明并统一对数底数。Equal distances on a logarithmic axis represent equal ratios, not equal differences. State and match the log base when comparing information or loss.
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