Modeling with Logarithmic Functions · 用对数函数建模
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| logarithmic model/ˌlɒɡəˈrɪθmɪk ˈmɒdl/ | 对数模型 | duì shù mó xíng |
| regression/rɪˈɡreʃn/ | 回归 | huí guī |
| parameters/pəˈræmɪtəz/ | 参数 | cān shù |
Growth that keeps slowing
- Some things improve fast at first, then crawl: a new skill, a filling reservoir, fading sound.
- The gains never stop, but each one is smaller than the last.
- That "fast then slow, forever" shape is exactly what a logarithm draws.
- So logarithmic models describe decelerating growth beautifully.
不断放慢的增长
- 有些东西起初进步很快,然后变得很慢:一项新技能、一个逐渐蓄满的水库、渐弱的声音。
- 收益从不停止,但每一次都比上一次更小。
- 那种"先快后慢,永不停止"的形状,正是对数所画出的。
- 所以对数模型能很好地描述减速的增长。
Building a logarithmic model
- A logarithmic model 对数模型 has the form $y = a\log x + c$.
- It shoots up steeply for small $x$, then rises ever more gently.
- Choose it when the data climbs quickly and then flattens without a ceiling.
- Two or three points can pin down the constants by hand.
构建对数模型
- 对数模型(logarithmic model)的形式是 $y = a\log x + c$。
- 对小的 $x$ 它陡然上升,然后越来越温和地升高。
- 当数据快速攀升、然后趋平却没有上限时,就选它。
- 两三个点就能手算确定这些常数。
A logarithmic model suits data that… · 对数模型适合……的数据。
Logs rise fast then flatten — perfect for growth that slows but never stops, like learning curves. · 对数先快升后趋平——非常适合放慢却不停止的增长,比如学习曲线。
Regression finds the fit
- With scattered data, run a logarithmic regression 回归 on a calculator.
- It returns the best-fit $a$ and $c$ for the curve $y = a\log x + c$.
- The curve threads the trend without matching any point exactly.
- The result is a formula ready to interpret and use.
回归找到拟合
- 面对散开的数据,在计算器上做一次对数回归(regression)。
- 它返回曲线 $y = a\log x + c$ 的最佳拟合 $a$ 和 $c$。
- 曲线穿过趋势,却不精确匹配任何一个点。
- 结果是一个可以解释和使用的公式。
A curve that rises fast, then flattens · 一条先快升、后趋平的曲线
y = a·log(x − b) + c
A logarithmic model climbs steeply at first, then slows without ever stopping — ideal for data that keeps slowing down. · 对数模型起初陡升,随后放慢却从不停止——非常适合不断放慢的数据。
Technology can find a logarithmic ____ that best fits a data set. · 技术可以找到最能拟合一组数据的对数____。
A logarithmic regression returns the $a$ and $c$ in · 入 $y = a\log x + c$ that fit the points best. · 对数回归返回 $y = a\log x + c$ 中最拟合这些点的 $a$ 和 $c$。
Select all · 所有 true statements about logarithmic models. · 选出关于对数模型的所有正确说法。
A log rises without bound — it slows but has no ceiling. The other three are correct. · 对数无界地上升——它放慢但没有上限。其余三条正确。
Interpret the parameters
- The parameters 参数 $a$ and $c$ carry real meaning.
- A larger $a$ stretches the curve, meaning a faster initial rise.
- The constant $c$ shifts the whole curve up or down to match the data's level.
- Reading these numbers turns the model back into a statement about the situation.
解释参数
- 参数(parameters)$a$ 和 $c$ 带有真实的含义。
- 更大的 $a$ 拉伸曲线,意味着更快的初始上升。
- 常数 $c$ 把整条曲线上下平移,以匹配数据的水平。
- 读懂这些数字,就把模型重新变回关于情境的陈述。
The parameters of a fitted model can be interpreted in terms of the real situation. · 拟合模型的参数可以用真实情境来解释。
Each constant carries meaning — a bigger $a$ means a faster initial rise, for example. · 每个常数都有含义——例如,更大的 $a$ 意味着更快的初始上升。
Which contrast is correct? · 下面哪个对比是正确的?
They are inverses: the exponential accelerates, while its mirror the log decelerates. · 它们互为反函数:指数在加速,而它的镜像对数在减速。
Log versus exponential contexts
- An exponential fits things that accelerate: populations, compound interest, viral spread.
- A logarithm fits things that decelerate: sensory perception, diminishing returns.
- They are inverses, so their model shapes are opposite.
- Ask "is growth speeding up or slowing down?" to pick the right family.
对数与指数的情境
- 指数适合加速的事物:人口、复利、病毒传播。
- 对数适合减速的事物:感官知觉、边际收益递减。
- 它们互为反函数,所以模型形状相反。
- 问"增长在加速还是减速?"来挑选正确的族。
A logarithmic model slows down but never stops — it has no upper limit. Do not use it for something that truly saturates at a hard ceiling (a different model handles that). It only fits growth that keeps rising, just more and more gently.
对数模型放慢却从不停止——它没有上限。不要把它用于真正在硬上限处饱和的事物(那要用另一种模型)。它只拟合持续上升、只是越来越温和的增长。
A study finds a test score rises with study hours as $S = 20\log_{10} h + 40$.
- At $h = 1$: $S = 20(0) + 40 = 40$.
- At $h = 10$: $S = 20(1) + 40 = 60$ — ten times the hours for $+20$ points.
- The gain from $10 \to 100$ hours is again just $+20$: clear diminishing returns.
一项研究发现测验分数随学习时数变化为 $S = 20\log_{10} h + 40$。
- 在 $h = 1$:$S = 20(0) + 40 = 40$。
- 在 $h = 10$:$S = 20(1) + 40 = 60$——十倍的时数换来 $+20$ 分。
- 从 $10 \to 100$ 小时的收益又只是 $+20$:明显的边际递减。
A logarithmic model $y = a\log x + c$ fits data that rises quickly then keeps slowing. Find it with a logarithmic regression, interpret its parameters in context, and remember it has no ceiling — it decelerates but never stops rising.
对数模型 $y = a\log x + c$ 拟合先快升、后不断放慢的数据。用对数回归找到它,在情境中解释它的参数,并记住它没有上限——它减速却从不停止上升。