Exponentials and logarithms · 指数函数与对数
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| exponential function/ˌekspəˈnenʃl ˈfʌŋkʃn/ | 指数函数 | zhǐ shù hán shù |
| exponent/ekˈspəʊnənt/ | 指数 | zhǐ shù |
| natural logarithms/ˈnætʃərəl ˈlɒɡərɪθəmz/ | 自然对数 | zì rán duì shù |
| compound interest/ˈkɒmpaʊnd ˈɪntrest/ | 复利 | fù lì |
| exponential decay/ˌekspəˈnenʃl dɪˈkeɪ/ | 指数衰减 | zhǐ shù shuāi jiǎn |
Repeated multiplication creates a model
- An exponential function 指数函数 such as $y=a^x$ has the variable in the exponent 指数, with $a>0$ and $a\ne1$.
- Its values grow for $a>1$ and decay for $0
重复乘法建立模型
- 指数函数 指数函数如 $y=a^x$ 的变量位于指数 指数中，其中 $a>0$ 和 $a\ne1$。
- 当 $a>1$ 时其值增长，当 $0
Exponential growth against a straight line · 指数增长对比直线
$y = a^x$
A constant factor always overtakes a constant amount, however steep the line. · 无论直线多陡，恒定因子终将超过恒定增量。
Use a logarithm as an inverse
- $\log_a y=x$ means $a^x=y$. Real logarithms need a positive argument and a positive base different from 1.
- Natural logarithms 自然对数 use base e. The graph $y=\ln x$ passes through $(1,0)$ and has vertical asymptote $x=0$; exponential graphs have y-intercept 1 and horizontal asymptote $y=0$.
使用对数作为逆运算
- $\log_a y=x$ 意味着 $a^x=y$。实对数需要正真数和不等于1的正底数。
- 自然对数 自然对数使用底数e。图像 $y=\ln x$ 经过 $(1,0)$ 并有垂直渐近线 $x=0$；指数图像有y轴截距1和水平渐近线 $y=0$。
What is log₁₀ 1000? · log₁₀ 1000 等于多少？
It asks what power of 10 gives 1000, and 10³ = 1000. · 它问 10 的几次方等于 1000，而 10³ = 1000。
Apply laws only within their domain
- For positive m and n with the same valid base, $\log(mn)=\log m+\log n$ and $\log(m/n)=\log m-\log n$.
- The power law $\log(m^k)=k\log m$ brings an exponent into a product. There is no corresponding law splitting the logarithm of a sum.
仅在定义域内应用定律
- 对于具有相同有效底数的正数m和n，$\log(mn)=\log m+\log n$ 和 $\log(m/n)=\log m-\log n$。
- 幂法则 $\log(m^k)=k\log m$ 将指数引入乘积中。不存在对应的法则可以将和的对数拆分。
Which law lets you solve for an unknown that sits in an exponent? · 哪条法则可用来求解指数中的未知数？
The power law brings k down to the front, where it can be divided out. The fourth option is not a law at all. · 幂法则将 k 移至前方以便约去。第四个选项根本不是法则。
log(m + n) equals log m + log n. · log(m + n) 不等于 log m + log n。
The addition law applies to a product inside the logarithm, never to a sum. · 加法法则适用于对数内的乘积，绝不适用于和。
State the time and financial assumptions
- Compound interest 复利 uses $A=P(1+r)^n$ for a stated rate per period and no other cash flows. The rate r is a fraction, not the percentage numeral.
- Exponential decay 指数衰减 uses a factor between 0 and 1. A continuous-time threshold can differ from the first qualifying whole-period sample; real processes need evidence before adopting a model.
陈述时间和财务假设
- 复利 复利使用 $A=P(1+r)^n$ 来表示每个周期的指定利率，且无其他现金流。利率 r 是一个分数，而非百分比数值。
- 指数衰减 指数衰减使用介于 0 和 1 之间的因子。连续时间的阈值可能与第一个符合条件的完整周期样本不同；真实过程在采用模型前需要证据支持。
5000 grows at 4% a year. After how many WHOLE years does it first exceed 7000? · 5000 以每年 4% 增长。经过多少个整年后首次超过 7000？
n > log 1.4 ÷ log 1.04 = 8.58, and years are whole, so the answer is 9. · n > log 1.4 ÷ log 1.04 = 8.58，年数为整数，故答案为 9。
Which base gives exponential decay? · 哪个底数会导致指数衰减？
A base between 0 and 1 shrinks the value at every step. A base of 1 never changes it. · 介于 0 和 1 之间的底数会使每一步的值缩小。底数为 1 则永不变化。
In one English sentence, say when you would use a logarithm to solve an equation. · 用一句英语句子说明何时会用对数解方程。
Example: "I take logarithms when the unknown is in the exponent, because the power law brings it down." · 示例：“当未知数位于指数中时，我取对数，因为幂法则能将其移下。”
A whole-year threshold. With $A(n)=500(1.04)^n$, the condition $A(n)>600$ gives $n>\ln(600/500)/\ln1.04\approx4.6486$. The first whole year is 5. Check years 4 and 5 to verify the strict inequality.
全年阈值。 使用 $A(n)=500(1.04)^n$ 时，条件 $A(n)>600$ 给出 $n>\ln(600/500)/\ln1.04\approx4.6486$。第一个完整年份是 5。请检查第 4 年和第 5 年以验证严格不等式。
Check equation and sketch domains separately. In $\ln(x-1)+\ln2=\ln10$, the equation needs $x>1$ and has solution 6. The separate basic graph $y=\ln x$ is defined for $x>0$. Sheet 1.6 practises both equations and graph features.
分别检查方程与草图定义域。 在 $\ln(x-1)+\ln2=\ln10$ 中，方程需要满足 $x>1$ 并得出解 6。独立的基图 $y=\ln x$ 的定义域为 $x>0$。练习纸 1.6 同时练习这两种方程与图形特征。
A positive exponential decay model stays above zero at every finite time. A physical quantity measured as zero may reflect detection limits or failure of the model, rather than an exact zero predicted by the formula.
一个正指数衰减模型在任何有限时间点都保持在零以上。若某物理量测量值为零，这可能反映了探测极限或模型失效，而非公式预测的确切零值。