Manipulating Logarithmic Expressions · 对数表达式的运算
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| quotient rule/ˈkwəʊʃənt ruːl/ | 除法法则 | chú fǎ fǎ zé |
| product rule/ˈprɒdʌkt ruːl/ | 乘法法则 | chéng fǎ fǎ zé |
| power rule/ˈpaʊə ruːl/ | 幂法则 | mì fǎ zé |
| change-of-base/tʃeɪndʒ ɒv beɪs/ | 换底 | huàn dǐ |
| equivalent/ɪˈkwɪvələnt/ | 等价 | děng jià |
Turning products into sums
- Before calculators, people multiplied huge numbers by adding their logarithms.
- That magic works because a log turns multiplication into addition.
- Three simple laws let you expand, condense, or re-base any log expression.
- They are just the exponent rules, viewed through the log's mirror.
把积变成和
- 在计算器出现之前,人们通过相加对数来把巨大的数相乘。
- 这个魔法之所以奏效,是因为对数把乘法变成了加法。
- 三条简单的法则让你能展开、合并或换底任何对数表达式。
- 它们不过是从对数的镜子里看到的指数法则。
Product and quotient rules
- The product rule 乘法法则 splits a product: $\log_b(xy) = \log_b x + \log_b y$.
- The quotient rule 除法法则 splits a quotient: $\log_b\!\left(\tfrac{x}{y}\right) = \log_b x - \log_b y$.
- Multiplication inside becomes addition outside; division becomes subtraction.
- These mirror "multiply powers → add exponents" and "divide → subtract".
乘法法则与除法法则
- 乘法法则(product rule)拆开一个积:$\log_b(xy) = \log_b x + \log_b y$。
- 除法法则(quotient rule)拆开一个商:$\log_b\!\left(\tfrac{x}{y}\right) = \log_b x - \log_b y$。
- 里面的乘法变成外面的加法;除法变成减法。
- 它们映照着"幂相乘 → 指数相加"和"相除 → 相减"。
The logarithm behind the laws · 法则背后的对数
y = a·log(x − b) + c
These laws come from the exponent rules in reverse. Explore the log curve whose algebra they describe. · 这些法则来自反过来的指数法则。探索它们所描述的这条对数曲线。
The product rule for logs says $\log_b(xy) =$ · 对数的乘法法则说 $\log_b(xy) =$
A log of a product is a sum · 和 of logs — the mirror of "multiplying powers adds exponents". · 积的对数是对数的和——是"幂相乘、指数相加"的镜像。
The quotient rule says $\log_b\!\left(\dfrac{x}{y}\right) = \log_b x - \log_b y$. · 除法法则说 $\log_b\!\left(\dfrac{x}{y}\right) = \log_b x - \log_b y$。
A log of a quotient is a difference of logs — the mirror of "dividing powers subtracts exponents". · 商的对数是对数的差——是"幂相除、指数相减"的镜像。
The power rule
- The power rule 幂法则 brings an exponent out front: $\log_b(x^n) = n\log_b x$.
- So $\log_2(8^3) = 3\log_2 8 = 3 \cdot 3 = 9$.
- A power buried inside a log becomes a simple multiplier.
- Together, the three rules can fully expand or condense a log expression.
幂法则
- 幂法则(power rule)把指数提到前面:$\log_b(x^n) = n\log_b x$。
- 所以 $\log_2(8^3) = 3\log_2 8 = 3 \cdot 3 = 9$。
- 藏在对数里的幂变成一个简单的倍数。
- 三条法则合在一起,可以完全展开或合并一个对数表达式。
The power rule for logs says $\log_b(x^n) =$ · 对数的幂法则说 $\log_b(x^n) =$
A power inside a log comes out front as a multiplier: $\log_b(x^n) = n\log_b x$. · 对数里面的幂可以提到前面作为倍数:$\log_b(x^n) = n\log_b x$。
Change of base
- Calculators only have $\log$ (base 10) and $\ln$ (base $e$) buttons.
- The change-of-base 换底 formula handles any other base: $\log_b x = \dfrac{\log x}{\log b}$.
- So $\log_2 7 = \dfrac{\log 7}{\log 2} \approx 2.807$.
- Pick whichever base your tools support, then divide.
换底
- 计算器只有 $\log$(以 10 为底)和 $\ln$(以 $e$ 为底)的按钮。
- 换底(change-of-base)公式处理任何其他底数:$\log_b x = \dfrac{\log x}{\log b}$。
- 所以 $\log_2 7 = \dfrac{\log 7}{\log 2} \approx 2.807$。
- 挑一个你工具支持的底数,然后相除。
The ____ formula lets you compute $\log_2 7$ as $\dfrac{\log 7}{\log 2}$ on a calculator. · ____公式让你在计算器上把 $\log_2 7$ 算成 $\dfrac{\log 7}{\log 2}$。
The change-of-base formula $\log_b x = \dfrac{\log x}{\log b}$ works with any convenient base. · 换底公式 $\log_b x = \dfrac{\log x}{\log b}$ 可用任何方便的底数。
Expand and condense
- Expanding uses the rules left-to-right to break one log into many.
- Condensing runs them right-to-left to combine many logs into one.
- Two expressions are equivalent 等价 if the rules convert one into the other.
- Choose the form that best fits the problem — solving usually wants a single log.
展开与合并
- 展开从左到右使用法则,把一个对数拆成许多个。
- 合并从右到左运行,把许多对数合成一个。
- 如果法则能把一个式子转换成另一个,那么两者是等价(equivalent)的。
- 选择最适合问题的形式——求解通常需要一个单独的对数。
Select all · 所有 correct log laws. · 选出所有正确的对数法则。
There is no · 否 rule for $\log(x+y)$ — the laws apply to products, quotients, and powers, not sums. The other three are correct. · $\log(x+y)$ 没有法则——这些法则适用于积、商和幂,而不是和。其余三条正确。
There is no rule for the log of a sum: $\log(x + y)$ does not equal $\log x + \log y$. The laws only apply to products, quotients, and powers inside the log. Splitting a sum is one of the most common log mistakes.
和的对数没有法则:$\log(x + y)$ 不等于 $\log x + \log y$。这些法则只适用于对数内部的积、商和幂。拆开一个和,是最常见的对数错误之一。
Condense $2\log x + \log y - \log z$ into one logarithm.
- Power rule: $2\log x = \log x^2$.
- Product then quotient: $\log x^2 + \log y - \log z = \log\!\left(\dfrac{x^2 y}{z}\right)$.
- One clean logarithm, ready to solve or evaluate.
把 $2\log x + \log y - \log z$ 合并成一个对数。
- 幂法则:$2\log x = \log x^2$。
- 先乘后商:$\log x^2 + \log y - \log z = \log\!\left(\dfrac{x^2 y}{z}\right)$。
- 一个干净的对数,可以直接求解或求值。
The log laws mirror the exponent rules: the product rule makes a sum, the quotient rule a difference, and the power rule a multiplier. The change-of-base formula evaluates any base with a calculator, and the rules show when two forms are equivalent — but never split $\log(x+y)$.
对数法则映照着指数法则:乘法法则得到和,除法法则得到差,幂法则得到倍数。换底公式让你用计算器求任何底数,法则也能显示两种形式何时等价——但绝不要拆开 $\log(x+y)$。