Semi-log Plots · 半对数图
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| exponential/ˌekspəˈnenʃl/ | 指数 | zhǐ shù |
| semi-log plot/ˈsemi lɒɡ plɒt/ | 半对数图 | bàn duì shù tú |
| logarithmic scale/ˌlɒɡəˈrɪθmɪk skeɪl/ | 对数刻度 | duì shù kè dù |
| slope/sləʊp/ | 斜率 | xié lǜ |
Straightening a curve
- A straight line is the easiest graph to read, fit, and extend.
- So scientists play a clever trick: re-scale an axis to turn a curve into a line.
- For exponential data, log-scaling the vertical axis does exactly that.
- The result is a semi-log plot, and it reveals hidden exponential behavior at a glance.
把曲线拉直
- 直线是最容易读、最容易拟合、最容易延伸的图。
- 所以科学家玩了个巧妙的把戏:重新缩放一个轴,把曲线变成直线。
- 对指数数据来说,把纵轴改成对数刻度正好做到这一点。
- 结果就是半对数图,它让隐藏的指数行为一眼可见。
What a semi-log plot is
- A semi-log plot 半对数图 uses a logarithmic scale 对数刻度 on one axis, usually the vertical.
- On a log scale, equal spaces mean equal multiplications: $1, 10, 100, 1000$ are evenly spaced.
- The horizontal axis stays ordinary (linear).
- "Semi" means only half the plot — one axis — is logarithmic.
什么是半对数图
- 半对数图(semi-log plot)在一个轴上使用对数刻度(logarithmic scale),通常是纵轴。
- 在对数刻度上,相等的间距意味着相等的乘法:$1, 10, 100, 1000$ 是等距的。
- 横轴保持普通(线性)。
- "半"意味着只有一半的图——一个轴——是对数的。
The exponential that a semi-log plot straightens · 被半对数图拉直的那条指数函数
y = a·bˣ
This exponential curves upward on ordinary axes. Re-scale the vertical axis logarithmically and it becomes a perfect straight line. · 这条指数函数在普通坐标轴上向上弯。把纵轴改成对数刻度,它就变成一条完美的直线。
A semi-log plot uses a logarithmic scale on… · 半对数图在……使用对数刻度。
"Semi" means half: a semi-log plot log-scales just one axis, most often the vertical one. · "半"意味着一半:半对数图只对一个轴用对数刻度,最常见是纵轴。
Why exponentials become straight
- Take an exponential 指数 $y = a\,b^x$ and apply a log to both sides.
- You get $\log y = \log a + x\log b$ — a linear equation in $x$.
- So plotting $\log y$ against $x$ gives a perfectly straight line.
- The log scale does the "$\log y$" for you automatically.
为什么指数变成直线
- 取一个指数(exponential)$y = a\,b^x$,对两边取对数。
- 你得到 $\log y = \log a + x\log b$——一个关于 $x$ 的线性方程。
- 所以把 $\log y$ 对 $x$ 作图,得到一条完美的直线。
- 对数刻度自动替你做了"$\log y$"这一步。
An exponential · 指数 relationship appears as a straight line on a semi-log plot. · 指数关系在半对数图上表现为一条直线。
Taking $\log$ of $y = a\,b^x$ gives $\log y = \log a + x\log b$ — linear in $x$, so the plot is straight. · 对 $y = a\,b^x$ 取 $\log$ 得到 $\log y = \log a + x\log b$——关于 $x$ 是线性的,所以图是直线。
Select all · 所有 true statements about semi-log plots. · 选出关于半对数图的所有正确说法。
Only exponential · 指数 data straightens on a semi-log plot — not every data set. The other three are correct. · 只有指数数据在半对数图上被拉直——并非每组数据。其余三条正确。
Reading slope and intercept
- The straight line's slope 斜率 equals $\log b$ — bigger slope, faster growth.
- Its intercept equals $\log a$ — that recovers the initial value $a$.
- So you can read both numbers of the exponential straight off the line.
- Fitting a line is far easier and more accurate than fitting a curve.
读斜率与截距
- 直线的斜率(slope)等于 $\log b$——斜率越大,增长越快。
- 它的截距等于 $\log a$——由此恢复初始值 $a$。
- 所以你能直接从直线上读出指数函数的两个数。
- 拟合一条直线比拟合一条曲线容易得多,也准确得多。
On a semi-log plot of an exponential, the ____ of the straight line equals $\log b$ (the growth rate). · 在指数函数的半对数图上,直线的____等于 $\log b$(增长率)。
From $\log y = \log a + x\log b$, the slope · 斜率 is $\log b$ and the intercept is $\log a$. · 由 $\log y = \log a + x\log b$,斜率是 $\log b$,截距是 $\log a$。
If data plots as a straight line on a semi-log graph, the best model is… · 如果数据在半对数图上画成一条直线,最佳模型是……
Straight on a semi-log plot is the signature of an exponential · 指数 relationship in the original data. · 在半对数图上呈直线,正是原始数据中指数关系的标志。
Building a model from the plot
- Plot your data on semi-log axes and see if it falls on a line.
- If it does, the underlying relationship is exponential.
- Measure the slope and intercept, then convert back to $a$ and $b$.
- This is a standard scientific tool for spotting exponential growth or decay.
从图上构建模型
- 把你的数据画在半对数坐标上,看它是否落在一条直线上。
- 如果是,背后的关系就是指数的。
- 量出斜率和截距,再换算回 $a$ 和 $b$。
- 这是一种发现指数增长或衰减的标准科学工具。
Only exponential data straightens on a semi-log plot — it is not a magic wand for every curve. If the points still bend on a semi-log plot, the relationship is not exponential, and you should try a different model.
只有指数数据在半对数图上被拉直——它不是对每条曲线都灵的魔杖。如果这些点在半对数图上仍然弯曲,那么关系不是指数的,你应该试试别的模型。
Bacteria counts double each hour: $1, 2, 4, 8, 16, \dots$
- On ordinary axes these curve upward steeply.
- On a semi-log plot ($\log$ of the count vs time) they fall on a straight line.
- The constant slope confirms a constant growth factor — genuine exponential growth.
细菌数量每小时翻一倍:$1, 2, 4, 8, 16, \dots$
- 在普通坐标轴上,这些点陡然向上弯。
- 在半对数图上(计数的 $\log$ 对时间),它们落在一条直线上。
- 恒定的斜率证实了恒定的增长因子——真正的指数增长。
A semi-log plot puts a logarithmic scale on one axis. An exponential relationship $y = a\,b^x$ becomes a straight line, because $\log y = \log a + x\log b$. The line's slope gives the growth rate and its intercept gives the initial value — a quick test for exponential data.
半对数图在一个轴上使用对数刻度。指数关系 $y = a\,b^x$ 变成一条直线,因为 $\log y = \log a + x\log b$。直线的斜率给出增长率,截距给出初始值——这是检验指数数据的快捷方法。