Exponentials and logarithms
| English | Chinese | Pinyin |
|---|---|---|
| exponential function | 指数函数 | zhǐ shù hán shù |
| exponent | 指数 | zhǐ shù |
| natural logarithms | 自然对数 | zì rán duì shù |
| compound interest | 复利 | fù lì |
| exponential decay | 指数衰减 | zhǐ shù shuāi jiǎn |
Growth that beats any straight line
- Add 100 yuan a month and your savings grow along a line. Multiply by 1.04 a year and they curve upward.
- An exponential function 指数函数 is $y = a^x$: the variable sits in the exponent 指数.
- It grows by a constant factor per step, which is why it eventually passes any straight line, however steep.
Exponential growth against a straight line
$y = a^x$
A constant factor always overtakes a constant amount, however steep the line.
A logarithm is the reverse question
- $\log_a y = x$ means exactly $a^x = y$. So $\log_{10} 1000 = 3$.
- It answers "what power?", which is the question you have whenever the unknown is upstairs.
- Natural logarithms 自然对数 use base $e \approx 2.718$ and are written $\ln$.
What is log₁₀ 1000?
It asks what power of 10 gives 1000, and 10³ = 1000.
The three laws
- The third law is the useful one: it brings a power down to the front, where you can solve for it.
- The first two turn multiplication into addition, which is what logarithms were invented for.
Which law lets you solve for an unknown that sits in an exponent?
The power law brings k down to the front, where it can be divided out. The fourth option is not a law at all.
log(m + n) equals log m + log n.
The addition law applies to a product inside the logarithm, never to a sum.
5000 yuan is invested at 4% a year. After how many whole years does it exceed 7000?
$5000(1.04)^n > 7000 \Rightarrow 1.04^n > 1.4$
Take logarithms of both sides: $n\log 1.04 > \log 1.4$, so $n > \dfrac{\log 1.4}{\log 1.04} = 8.58$.
It takes 9 whole years. Note the last step: the answer 8.58 is not the answer, because you cannot have 8.58 whole years.
5000 grows at 4% a year. After how many WHOLE years does it first exceed 7000?
n > log 1.4 ÷ log 1.04 = 8.58, and years are whole, so the answer is 9.
Decay, and where this is used
- Compound interest 复利 is $A = P(1+r)^n$ — exponential growth with $a > 1$.
- Exponential decay 指数衰减 has $0 < a < 1$: cooling coffee, medicine leaving the blood, radioactive material.
- These two shapes are why unit 6 sits under Mathematics I and reappears in the finance unit of GAC010.
Which base gives exponential decay?
A base between 0 and 1 shrinks the value at every step. A base of 1 never changes it.
In one English sentence, say when you would use a logarithm to solve an equation.
Example: "I take logarithms when the unknown is in the exponent, because the power law brings it down."
Reach for a logarithm the moment the unknown is an exponent. That single trigger covers almost every exam question in this unit — interest, decay, population, half-life.
$\log(m + n)$ is not $\log m + \log n$. The addition law works on a product inside the logarithm, not a sum. Writing the law out beside your working, once, prevents the most common error in the unit.