Inverses of Exponential Functions · 指数函数的反函数
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| inverse/ɪnˈvɜːs/ | 反函数 | fǎn hán shù |
| logarithmic function/ˌlɒɡəˈrɪθmɪk ˈfʌŋkʃn/ | 对数函数 | duì shù hán shù |
| domain/dəˈmeɪn/ | 定义域 | dìng yì yù |
| vertical asymptote/ˈvɜːtɪkl ˈæsɪmptəʊt/ | 竖直渐近线 | shù zhí jiàn jìn xiàn |
The exponential's mirror twin
- Every exponential function has a partner that runs it backward.
- Where $2^x$ turns an exponent into a value, its partner turns a value back into the exponent.
- That partner is the logarithm — and geometrically it is a perfect mirror image.
- Understanding the pair together makes both far easier.
指数函数的镜像双胞胎
- 每个指数函数都有一个把它倒着运行的搭档。
- $2^x$ 把指数变成一个值,而它的搭档把一个值变回指数。
- 那个搭档就是对数——从几何上看,它是一个完美的镜像。
- 把这一对放在一起理解,会让两者都容易得多。
Log is the inverse of exponential
- The logarithmic function 对数函数 $\log_b x$ is the inverse 反函数 of the exponential $b^x$.
- They share the same base and undo each other's work.
- $b^x$ takes an exponent to a value; $\log_b x$ takes the value back to the exponent.
- So logs and exponentials are two views of one relationship.
对数是指数的反函数
- 对数函数(logarithmic function)$\log_b x$ 是指数函数 $b^x$ 的反函数(inverse)。
- 它们共享同一个底数,并撤销彼此的工作。
- $b^x$ 把指数变成值;$\log_b x$ 把值变回指数。
- 所以对数和指数是同一种关系的两种视角。
The logarithmic function $\log_b x$ is the ____ of the exponential function $b^x$. · 对数函数 $\log_b x$ 是指数函数 $b^x$ 的____。
They undo each other, so the log function is the inverse · 反函数 of the exponential with the same base. · 它们互相撤销,所以对数函数是同底数指数函数的反函数。
Reflection across y = x
- Because they are inverses, their graphs are reflections across the line $y = x$.
- The soaring exponential becomes a slowly-rising logarithm, and vice versa.
- Every point $(a, b)$ on $b^x$ corresponds to $(b, a)$ on $\log_b x$.
- Seeing one graph, you can sketch the other by flipping across $y = x$.
关于 y = x 的反射
- 因为它们互为反函数,它们的图像关于直线 $y = x$ 反射。
- 飞升的指数变成缓慢上升的对数,反之亦然。
- $b^x$ 上的每个点 $(a, b)$ 对应 $\log_b x$ 上的 $(b, a)$。
- 看到一条图像,你就能通过关于 $y = x$ 翻转来画出另一条。
The exponential curve whose mirror is the logarithm · 其镜像是对数的指数曲线
y = 2ˣ
Picture reflecting this curve y = 2ˣ across the line y = x. The mirror image is the graph of y = log₂ x. · 想象把这条曲线 y = 2ˣ 关于直线 y = x 反射。镜像就是 y = log₂ x 的图像。
Swapped domain, range, and asymptote
- Inverses swap domain and range. The exponential's range $y > 0$ becomes the log's domain 定义域 $x > 0$.
- The exponential's horizontal asymptote $y = 0$ becomes the log's vertical asymptote 竖直渐近线 $x = 0$.
- The log's range is all real numbers — it can be any exponent.
- So the log function exists only for positive inputs, hugging the y-axis on the left.
交换的定义域、值域与渐近线
- 反函数交换定义域和值域。指数函数的值域 $y > 0$ 变成对数的定义域(domain)$x > 0$。
- 指数函数的水平渐近线 $y = 0$ 变成对数的竖直渐近线(vertical asymptote)$x = 0$。
- 对数的值域是全体实数——它可以是任何指数。
- 所以对数函数只对正的输入存在,在左侧紧贴 y 轴。
The exponential has a horizontal asymptote; its inverse, the log, has a ____ asymptote instead. · 指数函数有一条水平渐近线;它的反函数——对数——则有一条____渐近线。
Reflecting across $y=x$ turns the horizontal asymptote $y=0$ into the vertical asymptote $x=0$. · 关于 $y=x$ 反射,把水平渐近线 $y=0$ 变成竖直渐近线 $x=0$。
The domain of $\log_b x$ is $x > 0$, which was the range of the exponential. · $\log_b x$ 的定义域是 $x > 0$,这正是指数函数的值域。
Inverses swap domain and range: the exponential's range ($y>0$) becomes the log's domain ($x>0$). · 反函数交换定义域和值域:指数函数的值域($y>0$)变成对数的定义域($x>0$)。
Select all · 所有 true statements about the exponential and its inverse. · 选出关于指数函数及其反函数的所有正确说法。
They are inverses, not the same function. The other three correctly describe the relationship. · 它们是反函数,而不是同一个函数。其余三条正确地描述了这种关系。
They undo each other
- Composing a function with its inverse cancels: $\log_b(b^x) = x$ and $b^{\log_b x} = x$.
- So $\log_2(2^7) = 7$ and $10^{\log_{10} 5} = 5$.
- This cancelling is exactly how logs solve equations with the variable in the exponent.
- Apply the log to both sides, and the exponent comes down where you can reach it.
它们互相撤销
- 把一个函数与它的反函数复合会抵消:$\log_b(b^x) = x$ 且 $b^{\log_b x} = x$。
- 所以 $\log_2(2^7) = 7$,$10^{\log_{10} 5} = 5$。
- 这种抵消正是对数求解变量在指数上的方程的方法。
- 对两边取对数,指数就落到你够得着的地方。
What is $\log_2\!\left(2^5\right)$? · $\log_2\!\left(2^5\right)$ 是多少?
The log undoes the exponential: $\log_2(2^5) = 5$. Inverse functions cancel. · 对数撤销指数:$\log_2(2^5) = 5$。反函数互相抵消。
The exponential accepts any real input but only outputs positives; the log is the mirror — it only accepts positive inputs but outputs any real. Do not give a logarithm a zero or negative argument, and do not expect an exponential to ever output one.
指数函数接受任何实数输入,却只输出正数;对数是镜像——它只接受正的输入,却输出任何实数。不要给对数一个零或负的自变量,也不要指望指数函数会输出一个。
Use the inverse relationship to solve $10^x = 1000$.
- Take $\log_{10}$ of both sides: $\log_{10}(10^x) = \log_{10} 1000$.
- The left side cancels to $x$; the right side is $3$ (since $10^3 = 1000$).
- So $x = 3$ — the logarithm undid the exponential.
用反函数关系求解 $10^x = 1000$。
- 对两边取 $\log_{10}$:$\log_{10}(10^x) = \log_{10} 1000$。
- 左边抵消为 $x$;右边是 $3$(因为 $10^3 = 1000$)。
- 所以 $x = 3$——对数撤销了指数。
The logarithmic function is the inverse of the exponential with the same base: their graphs reflect across $y = x$. Domain and range swap, so the log has domain $x > 0$ and a vertical asymptote at $x = 0$. They undo each other, which is how logs solve exponential equations.
对数函数是同底数指数函数的反函数:它们的图像关于 $y = x$ 反射。定义域和值域互换,所以对数的定义域是 $x > 0$,并在 $x = 0$ 处有一条竖直渐近线。它们互相撤销,这正是对数求解指数方程的方法。