Logarithmic Functions · 对数函数
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| logarithmic function/ˌlɒɡəˈrɪθmɪk ˈfʌŋkʃn/ | 对数函数 | duì shù hán shù |
| base/beɪs/ | 底数 | dǐ shù |
| domain/dəˈmeɪn/ | 定义域 | dìng yì yù |
| vertical asymptote/ˈvɜːtɪkl ˈæsɪmptəʊt/ | 竖直渐近线 | shù zhí jiàn jìn xiàn |
| end behavior/end bɪˈheɪvjə/ | 末端行为 | mò duān xíng wéi |
The slow-and-steady curve
- The exponential rockets upward; its mirror, the logarithm, does the opposite.
- A logarithm rises forever but slower and slower, never flattening to a stop.
- It clings to the y-axis on the left and creeps upward on the right.
- This gentle shape models sound loudness, acidity, and the "feel" of big numbers.
又慢又稳的曲线
- 指数向上飞升;它的镜像——对数——恰好相反。
- 对数永远上升,却越来越慢,从不停下变平。
- 它在左侧紧贴 y 轴,在右侧缓缓向上爬。
- 这种温和的形状用来给声音响度、酸度,以及大数字的"感觉"建模。
The graph of a logarithm
- A logarithmic function 对数函数 is $f(x) = \log_b x$, the inverse of $b^x$.
- Its graph passes through $(1, 0)$, since $\log_b 1 = 0$ for every base.
- To the right of $1$ it is positive; between $0$ and $1$ it is negative.
- Read the shape as the exponential's reflection across $y = x$.
对数的图像
- 对数函数(logarithmic function)是 $f(x) = \log_b x$,即 $b^x$ 的反函数。
- 它的图像经过 $(1, 0)$,因为对每个底数都有 $\log_b 1 = 0$。
- 在 $1$ 的右边它为正;在 $0$ 和 $1$ 之间它为负。
- 把这个形状读作指数关于 $y = x$ 的反射。
Shape the logarithmic curve · 塑造对数曲线
y = a·log(x − b) + c
This log curve climbs slowly and clings to the y-axis. It exists only for x greater than its vertical asymptote. · 这条对数曲线爬升缓慢,紧贴 y 轴。它只在 x 大于它的竖直渐近线处存在。
Domain and vertical asymptote
- The domain 定义域 is $x > 0$ — you cannot take the log of zero or a negative.
- As $x \to 0^+$, the output plunges to $-\infty$: a vertical asymptote 竖直渐近线 at $x = 0$.
- The graph never touches the y-axis, only hugs it more and more tightly.
- Its range, by contrast, is all real numbers.
定义域与竖直渐近线
- 定义域(domain)是 $x > 0$——你不能取零或负数的对数。
- 当 $x \to 0^+$ 时,输出俯冲向 $-\infty$:在 $x = 0$ 处有一条竖直渐近线(vertical asymptote)。
- 图像从不触碰 y 轴,只是越来越紧地贴着它。
- 相比之下,它的值域是全体实数。
The domain of $f(x) = \log_b x$ is… · $f(x) = \log_b x$ 的定义域是……
You can only take the log of a positive number, so the domain is $x > 0$. · 你只能取正数的对数,所以定义域是 $x > 0$。
A logarithmic function has a ____ asymptote at $x = 0$. · 对数函数在 $x = 0$ 处有一条____渐近线。
As $x \to 0^+$ the output dives to $-\infty$, so $x = 0$ is a vertical asymptote. · 当 $x \to 0^+$ 时输出俯冲向 $-\infty$,所以 $x = 0$ 是一条竖直渐近线。
The graph of $y = \log_b x$ crosses the x-axis at… · $y = \log_b x$ 的图像在……处穿过 x 轴。
Since $b^0 = 1$, we have $\log_b 1 = 0$: the x-intercept is always at $x = 1$. · 因为 $b^0 = 1$,所以 $\log_b 1 = 0$:x 轴交点总在 $x = 1$。
Select all · 所有 true statements about $y = \log_b x$ (with $b > 1$). · 选出关于 $y = \log_b x$($b > 1$)的所有正确说法。
It rises without bound, so it has no maximum. The other three are correct. · 它无限上升,所以没有最大值。其余三条正确。
The base sets the direction
- If the base 底数 $b > 1$, the log function is increasing (slowly rising).
- If $0 < b < 1$, it is decreasing instead.
- A bigger base makes the curve climb even more gently.
- Whatever the base, the curve still passes through $(1, 0)$ and has the same asymptote.
底数决定方向
- 如果底数(base)$b > 1$,对数函数是递增的(缓慢上升)。
- 如果 $0 < b < 1$,它则是递减的。
- 底数越大,曲线爬升得越温和。
- 无论底数如何,曲线仍经过 $(1, 0)$,并有相同的渐近线。
A logarithmic function keeps increasing forever, but more and more slowly. · 对数函数会永远递增,但越来越慢。
It rises without bound as $x \to \infty$, yet its slope flattens — the opposite of exponential speed-up. · 当 $x \to \infty$ 时它无限上升,但斜率变平——与指数的加速正好相反。
End behavior: unbounded but slow
- As $x \to \infty$, the end behavior 末端行为 is $\log_b x \to \infty$ — it never levels off.
- But it grows so slowly that reaching $y = 10$ can take an enormous $x$.
- Compared with an exponential's explosion, the logarithm barely moves.
- That is exactly why logs are used to compress huge ranges of numbers.
末端行为:无界但缓慢
- 当 $x \to \infty$ 时,末端行为(end behavior)是 $\log_b x \to \infty$——它从不趋平。
- 但它增长得如此之慢,以至于到达 $y = 10$ 可能需要一个巨大的 $x$。
- 与指数的爆炸相比,对数几乎不动。
- 这正是为什么对数被用来压缩巨大范围的数字。
A logarithm's slow growth is not the same as "having a ceiling". It keeps rising forever — there is no horizontal asymptote and no maximum. It just takes longer and longer to climb each additional unit.
对数的缓慢增长并不等于"有个上限"。它永远上升——没有水平渐近线,也没有最大值。只是每多爬一个单位所花的时间越来越长。
Describe $y = \log_2 x$ at a few points.
- $\log_2 1 = 0$ (the x-intercept), $\log_2 2 = 1$, $\log_2 4 = 2$, $\log_2 8 = 3$.
- To gain each $+1$ in output, the input must double — growth is slowing.
- Near $x = 0$ the curve dives down the vertical asymptote toward $-\infty$.
在几个点上描述 $y = \log_2 x$。
- $\log_2 1 = 0$(x 轴交点),$\log_2 2 = 1$,$\log_2 4 = 2$,$\log_2 8 = 3$。
- 输出每增加 $+1$,输入就必须翻倍——增长在放慢。
- 在 $x = 0$ 附近,曲线沿着竖直渐近线俯冲向 $-\infty$。
A logarithmic function $\log_b x$ has domain $x > 0$, a vertical asymptote at $x = 0$, and passes through $(1, 0)$. The base decides increasing vs decreasing, and its end behavior is unbounded but ever-slowing growth — the mirror of an exponential.
对数函数 $\log_b x$ 的定义域是 $x > 0$,在 $x = 0$ 处有竖直渐近线,并经过 $(1, 0)$。底数决定递增还是递减,它的末端行为是无界但不断放慢的增长——正是指数的镜像。