Exponential and Logarithmic Equations · 指数方程与对数方程
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| exponential equation/ˌekspəˈnenʃl ɪˈkweɪʒn/ | 指数方程 | zhǐ shù fāng chéng |
| logarithmic equation/ˌlɒɡəˈrɪθmɪk ɪˈkweɪʒn/ | 对数方程 | duì shù fāng chéng |
| extraneous/ekˈstreɪnɪəs/ | 增根 | zēng gēn |
Finding a hidden exponent
- "$2$ to what power equals $5$?" is an equation with the unknown up in the exponent.
- You cannot just divide it out — the variable is trapped inside a power.
- Logarithms are the tool that frees it, and exponential form frees log equations.
- Between these two moves, you can solve almost any exponential or log equation.
寻找隐藏的指数
- "$2$ 的几次方等于 $5$?"是一个未知量在指数上的方程。
- 你不能简单地把它除掉——变量被困在幂里面。
- 对数正是解放它的工具,而指数形式则解放对数方程。
- 在这两招之间,你几乎能解出任何指数或对数方程。
Solving exponential equations
- An exponential equation 指数方程 has the variable in an exponent, like $2^x = 8$.
- If both sides can share a base, match them: $2^x = 2^3$ gives $x = 3$.
- If the bases will not match, take a logarithm of both sides.
- The power rule then brings the exponent down where you can solve for it.
求解指数方程
- 指数方程(exponential equation)的变量在指数上,比如 $2^x = 8$。
- 如果两边能配成同底数,就配上:$2^x = 2^3$ 给出 $x = 3$。
- 如果底数无法配同,就对两边取对数。
- 幂法则随后把指数拉到你能求解的地方。
Find where the exponential reaches a target value · 找到指数函数达到目标值的地方
y = 2ˣ
Solving 2ˣ = 5 means finding the x where this curve hits height 5. A logarithm gives that x exactly. · 求解 2ˣ = 5 就是找到这条曲线达到高度 5 的那个 x。对数能精确给出那个 x。
Solve $2^x = 8$ by matching bases. · 用配同底数的方法求解 $2^x = 8$。
$8 = 2^3$, so $2^x = 2^3$ gives $x = 3$ — equal bases mean equal exponents. · $8 = 2^3$,所以 $2^x = 2^3$ 给出 $x = 3$——底数相等则指数相等。
To solve $2^x = 5$, where the bases will not match, you should… · 要求解 $2^x = 5$(底数无法配同),你应该……
Taking $\log$ of both sides brings the exponent down: $x = \log_2 5 \approx 2.32$. · 对两边取 $\log$ 把指数拉下来:$x = \log_2 5 \approx 2.32$。
Solving logarithmic equations
- A logarithmic equation 对数方程 has the variable inside a log, like $\log_2 x = 4$.
- Rewrite it in exponential form: $\log_2 x = 4$ means $x = 2^4 = 16$.
- If several logs appear, condense them into one first, then convert.
- Exponential form is the master key for any single-log equation.
求解对数方程
- 对数方程(logarithmic equation)的变量在对数里面,比如 $\log_2 x = 4$。
- 把它改写成指数形式:$\log_2 x = 4$ 意味着 $x = 2^4 = 16$。
- 如果出现好几个对数,先把它们合并成一个,再转换。
- 指数形式是任何单个对数方程的万能钥匙。
Solve the logarithmic equation $\log_2 x = 4$. · 求解对数方程 $\log_2 x = 4$。
Rewrite in exponential form: $x = 2^4 = 16$. · 改写成指数形式:$x = 2^4 = 16$。
Check for extraneous solutions
- Solving can produce a candidate that breaks the original equation's domain.
- A logarithm needs a positive argument, so any solution giving $\log(\text{negative})$ is rejected.
- Such a false answer is called an extraneous 增根 solution.
- Always substitute back and confirm every argument stays positive.
检查增根
- 求解可能产生一个破坏原方程定义域的候选解。
- 对数需要正的自变量,所以任何要对负数取对数的解都要舍弃。
- 这样一个错误的答案被称为增根(extraneous)。
- 永远要代回去,确认每个自变量都保持为正。
A candidate solution that makes a log argument negative must be rejected as ____. · 一个使对数的自变量变为负数的候选解,必须作为____而被舍弃。
An extraneous solution satisfies the rearranged equation but breaks the original domain — always check. · 增根满足变形后的方程,却破坏了原来的定义域——一定要检查。
Select all · 所有 true statements about solving these equations. · 选出关于求解这些方程的所有正确说法。
Some candidates are extraneous and must be rejected. The other three are sound strategies. · 有些候选解是增根,必须舍弃。其余三条是可靠的策略。
Inequalities
- Exponential and logarithmic functions are monotonic — always increasing or always decreasing.
- So an inequality like $2^x > 8$ solves the same way, keeping the direction: $x > 3$.
- For a decreasing base ($0 < b < 1$), the inequality sign flips.
- Solve as an equation first, then decide which side of the boundary satisfies it.
不等式
- 指数函数和对数函数是单调的——总是递增或总是递减。
- 所以像 $2^x > 8$ 这样的不等式用同样的方法解,方向不变:$x > 3$。
- 对于递减的底数($0 < b < 1$),不等号方向翻转。
- 先当方程解,再判断边界的哪一侧满足它。
Never forget to check log-equation answers. Solving $\log(x) + \log(x-3) = 1$ may hand you $x = 5$ and $x = -2$, but $x = -2$ makes $\log(-2)$ undefined — it is extraneous and must be thrown out.
永远不要忘了检查对数方程的答案。求解 $\log(x) + \log(x-3) = 1$ 可能给你 $x = 5$ 和 $x = -2$,但 $x = -2$ 会让 $\log(-2)$ 无定义——它是增根,必须扔掉。
Solve $2^x = 5$.
- Bases will not match, so take $\log_2$ of both sides.
- $\log_2(2^x) = \log_2 5$, and the left side simplifies to $x$.
- So $x = \log_2 5 \approx 2.32$ — the exponent the logarithm revealed.
求解 $2^x = 5$。
- 底数无法配同,所以对两边取 $\log_2$。
- $\log_2(2^x) = \log_2 5$,左边化简为 $x$。
- 所以 $x = \log_2 5 \approx 2.32$——对数揭示出的那个指数。
Solve an exponential equation by matching bases, or by taking a logarithm to bring the exponent down. Solve a logarithmic equation by rewriting it in exponential form. Always check for extraneous solutions, since a log's argument must stay positive.
通过配同底数,或取对数把指数拉下来,求解指数方程。通过改写成指数形式,求解对数方程。永远要检查增根,因为对数的自变量必须保持为正。