Logarithms · 对数
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| exponential/ˌekspəˈnenʃl/ | 指数 | zhǐ shù |
| logarithm/ˈlɒɡərɪθəm/ | 对数 | duì shù |
| base/beɪs/ | 底数 | dǐ shù |
| natural log/ˈnætʃərəl lɒɡ/ | 自然对数 | zì rán duì shù |
| common log/ˈkɒmən lɒɡ/ | 常用对数 | cháng yòng duì shù |
The reverse question
- Exponentials answer "start at 1 and multiply by 2 five times — what do you get?" ($2^5 = 32$).
- Logarithms answer the reverse: "how many times must I multiply by 2 to reach 32?" ($5$).
- That reverse question comes up everywhere: pH, earthquakes, decibels, compound interest.
- A logarithm is simply the exponent, pulled back out.
反过来的问题
- 指数回答"从 1 开始,乘以 2 五次——得到什么?"($2^5 = 32$)。
- 对数回答反过来的问题:"我必须乘以 2 多少次才能到达 32?"($5$)。
- 这个反过来的问题无处不在:pH 值、地震、分贝、复利。
- 对数不过就是被重新拉出来的那个指数。
A logarithm is an exponent
- A logarithm 对数 answers: to what power must the base 底数 be raised to get a value?
- $\log_b y = x$ means exactly $b^x = y$ — the log is the exponent $x$.
- So $\log_2 8 = 3$ because $2^3 = 8$.
- Reading a log as "the exponent that works" makes it far less mysterious.
对数就是指数
- 对数(logarithm)回答:底数(base)要升到几次方才能得到某个值?
- $\log_b y = x$ 恰好意味着 $b^x = y$——对数就是那个指数 $x$。
- 所以 $\log_2 8 = 3$,因为 $2^3 = 8$。
- 把对数读作"那个管用的指数",它就没那么神秘了。
Read a logarithm off the exponential curve · 从指数曲线上读出对数
y = 2ˣ
Find the y-value 8 on the curve y = 2ˣ. The x that produces it is the exponent — that is log base 2 of 8. · 在曲线 y = 2ˣ 上找到 y 值 8。产生它的那个 x 就是指数——那就是 2 为底 8 的对数。
The statement $\log_b y = x$ means the same as… · 式子 $\log_b y = x$ 的意思与……相同。
A logarithm is the exponent: $\log_b y = x$ asks "$b$ to what power gives $y$?", so $b^x = y$. · 对数就是指数:$\log_b y = x$ 问的是"$b$ 的几次方等于 $y$?",所以 $b^x = y$。
Evaluate · 评价 $\log_2 8$. · 求 $\log_2 8$。
$2^3 = 8$, so the exponent is $3$: $\log_2 8 = 3$. · $2^3 = 8$,所以指数是 $3$:$\log_2 8 = 3$。
Common and natural logs
- Two bases are so common they get shortcuts.
- The common log 常用对数 $\log$ (with no base shown) has base $10$.
- The natural log 自然对数 $\ln$ has base $e \approx 2.718$.
- Calculators have both buttons; other bases use the change-of-base formula.
常用对数与自然对数
- 有两个底数太常用了,于是有了简写。
- 常用对数(common log)$\log$(不写底数)以 $10$ 为底。
- 自然对数(natural log)$\ln$ 以 $e \approx 2.718$ 为底。
- 计算器上两个按钮都有;其他底数用换底公式。
The natural log $\ln$ uses which base? · 自然对数 $\ln$ 用的是哪个底数?
The natural log has base $e$; the common log ($\log$ with no base written) has base $10$. · 自然对数以 $e$ 为底;常用对数($\log$ 不写底数)以 $10$ 为底。
Select all · 所有 true statements about logarithms. · 选出关于对数的所有正确说法。
The argument of a log must be positive, so you cannot take the log of a negative. The other three are correct. · 对数的自变量必须为正,所以不能取负数的对数。其余三条正确。
Converting between forms
- Every exponential 指数 statement has a matching logarithmic one.
- $b^x = y \iff \log_b y = x$ — the same fact written two ways.
- Rewrite $10^3 = 1000$ as $\log_{10} 1000 = 3$.
- Switching forms is the key trick for solving equations with unknown exponents.
在两种形式之间转换
- 每一个指数(exponential)式子都有一个对应的对数式子。
- $b^x = y \iff \log_b y = x$——同一个事实的两种写法。
- 把 $10^3 = 1000$ 改写成 $\log_{10} 1000 = 3$。
- 切换形式是求解带未知指数方程的关键技巧。
The argument must be positive
- You can only take the logarithm of a positive number.
- A positive base raised to any power stays positive, so $b^x$ never reaches $0$ or a negative.
- Therefore $\log_b y$ is undefined for $y \le 0$.
- This restriction shapes the domain of every logarithmic function.
自变量必须为正
- 你只能取正数的对数。
- 正底数升到任何次幂都保持为正,所以 $b^x$ 永远到不了 $0$ 或负数。
- 因此当 $y \le 0$ 时 $\log_b y$ 无定义。
- 这个限制塑造了每一个对数函数的定义域。
The argument (input) of a logarithm must be , because a positive base to any power stays positive. · 对数的自变量(输入)必须是,因为正底数的任何次幂都保持为正。
Since $b^x > 0$ always, no exponent can produce zero or a negative — so $\log_b y$ needs $y > 0$. · 因为 $b^x > 0$ 恒成立,没有指数能产生零或负数——所以 $\log_b y$ 需要 $y > 0$。
$\log_b y$ is the exponent, not a product. $\log_2 8$ is $3$ (the power), not $2 \times 8$. And you can never take the log of $0$ or a negative number — the argument must be strictly positive.
$\log_b y$ 是那个指数,不是乘积。$\log_2 8$ 是 $3$(那个幂),不是 $2 \times 8$。而且你永远不能取 $0$ 或负数的对数——自变量必须严格为正。
Solve $2^x = 16$ using a logarithm.
- Rewrite in log form: $x = \log_2 16$.
- Ask "$2$ to what power is $16$?" — since $2^4 = 16$, the answer is $4$.
- So $x = 4$: the logarithm pulled the exponent out for us.
用对数求解 $2^x = 16$。
- 改写成对数形式:$x = \log_2 16$。
- 问"$2$ 的几次方是 $16$?"——因为 $2^4 = 16$,答案是 $4$。
- 所以 $x = 4$:对数替我们把指数拉了出来。
A logarithm is an exponent: $\log_b y = x$ means $b^x = y$. The common log uses base $10$ and the natural log uses base $e$. Every exponential statement converts to a log statement, and the argument of a log must always be positive.
对数是一个指数:$\log_b y = x$ 意味着 $b^x = y$。常用对数以 $10$ 为底,自然对数以 $e$ 为底。每一个指数式子都能转换成对数式子,而对数的自变量必须始终为正。