Logarithmic and exponential functions · 对数函数与指数函数
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| logarithm/ˈlɒɡərɪθəm/ | 对数 | duì shù |
| laws of logarithms/lɔːz ɒv ˈlɒɡərɪθəmz/ | 对数定律 | duì shù dìng lǜ |
| natural logarithm/ˈnætʃərəl ˈlɒɡərɪθəm/ | 自然对数 | zì rán duì shù |
| linear form/ˈlɪnɪə fɔːm/ | 线性形式 | xiàn xìng xíng shì |
The number that grows forever
- A bank account doubles every 10 years. After 100 years, it's grown by a factor of $2^{10} = 1024$.
- But what if you want to know: "After how many years will it be worth exactly 500 times the original?" That's a question only logarithms 对数 can answer.
永远增长的数
- 一个银行账户每 10 年翻一倍。100 年后,它增长了 $2^{10} = 1024$ 倍。
- 但如果你想知道:“多少年后它会值恰好原来的 500 倍?”那是只有对数(logarithms)才能回答的问题。
What is a logarithm?
- A logarithm answers "what power?": if $a^x = y$ then $x = \log_a y$.
- Example: $2^x = 32 \Rightarrow x = \log_2 32 = 5$.
Worked example. Solve $3^x = 81$. Since $81 = 3^4$, we get $x = \log_3 81 = 4$.
Earthquake strength is measured on the logarithmic Richter scale
什么是对数?
- 一个对数回答“什么幂?”:如果 $a^x = y$ 那么 $x = \log_a y$。
- 例子:$2^x = 32 \Rightarrow x = \log_2 32 = 5$。
算例。 解 $3^x = 81$。因为 $81 = 3^4$,我们得到 $x = \log_3 81 = 4$。
地震强度用对数的里氏震级测量
Exponential growth · 指数增长
y = a·bˣ
Change the base b: when b > 1 the curve grows, when 0 < b < 1 it decays · 衰变 — and it always passes through (0, a). · 改变底 b:当 b > 1 时曲线增长,当 0 < b < 1 时它衰减——而且它总是过 (0, a)。
Evaluate log₂(8). · 求 log₂(8)。
2³ = 8, so log₂(8) = 3. · 2³ = 8,所以 log₂(8) = 3。
Solve 2ˣ = 8 for x. · 解 2ˣ = 8,求 x。
2³ = 8, so x = 3 (or x = log₂8 = 3). · 2³ = 8,所以 x = 3(或 x = log₂8 = 3)。
The laws of logarithms 对数定律
- The laws of logarithms:
- $\log(mn) = \log m + \log n$,
- $\log\dfrac{m}{n} = \log m - \log n$,
- $\log(m^k) = k\log m$.
Logs don't distribute over addition. $\log(a + b) \neq \log a + \log b$. The laws only apply to products, quotients, and powers — not sums.
Populations can grow exponentially when resources are plentiful
对数法则
- 对数法则(laws of logarithms):
- $\log(mn) = \log m + \log n$,
- $\log\dfrac{m}{n} = \log m - \log n$,
- $\log(m^k) = k\log m$。
对数不对加法分配。 $\log(a + b) \neq \log a + \log b$。这些法则只适用于乘积、商和幂——不适用于和。
当资源充足时,种群能指数地增长
Which is a correct law of logarithms? · 哪个是正确的对数法则?
log(mn) = log m + log n; and log(m^k) = k log m. · log(mn) = log m + log n;而 log(m^k) = k log m。
log(a + b) = log a + log b. · log(a + b) = log a + log b。
Logarithms do not distribute over addition. log(a + b) ≠ log a + log b. · 对数不对加法分配。log(a + b) ≠ log a + log b。
e and ln
- $e^x$ and the natural logarithm 自然对数 $\ln x$ are inverses: $\ln(e^x) = x$ and $e^{\ln x} = x$.
- When the unknown is in the power, take logs of both sides.
$e^x$ and $\ln x$ are inverse functions: their graphs are reflections in the line $y = x$.
e 与 ln
- $e^x$ 和自然对数(natural logarithm)$\ln x$ 是互逆的:$\ln(e^x) = x$ 而 $e^{\ln x} = x$。
- 当未知数在幂里时,对两边取对数。
$e^x$ 和 $\ln x$ 是互逆函数:它们的图是在直线 $y = x$ 中的反射。
What is ln(e⁵)? · ln(e⁵) 是多少?
ln and e are inverses, so ln(e⁵) = 5. · ln 和 e 是互逆的,所以 ln(e⁵) = 5。
Linear form 线性形式 and modelling
- Linear form: $y = Ax^n$ becomes $\ln y = \ln A + n\ln x$ — a line with gradient $n$, intercept $\ln A$.
- This lets you find the power law from experimental data by plotting $\ln y$ vs $\ln x$.
- Logarithms and indices (powers) are reverse operations.
线性形式与建模
- 线性形式(linear form):$y = Ax^n$ 变成 $\ln y = \ln A + n\ln x$——一条斜率 $n$、截距 $\ln A$ 的直线。
- 这让你通过画 $\ln y$ 对 $\ln x$ 从实验数据找到幂律。
- 对数与指数(indices,幂)是互逆运算。
If y = 5x³, then ln y = ln 5 + n ln x. What is n? · 如果 y = 5x³,那么 ln y = ln 5 + n ln x。n 是多少?
ln(5x³) = ln 5 + 3 ln x, so n = 3 (the power becomes the gradient). · ln(5x³) = ln 5 + 3 ln x,所以 n = 3(幂变成斜率)。
You've got it
- a log answers "what power?"; laws: $\log mn = \log m + \log n$, $\log m^k = k\log m$
- $e^x$ and $\ln x$ are inverses; take logs when the unknown is a power
- linear form: plot $\ln y$ vs $\ln x$ → gradient $n$, intercept $\ln A$
你掌握了
- 一个对数回答“什么幂?”;法则:$\log mn = \log m + \log n$,$\log m^k = k\log m$
- $e^x$ 和 $\ln x$ 是互逆的;当未知数是一个幂时取对数
- 线性形式:画 $\ln y$ 对 $\ln x$ → 斜率 $n$,截距 $\ln A$