Modeling with Logarithmic Functions
Đã giới thiệu| English | Ý nghĩa |
|---|---|
| logarithmic model/ˌlɒɡəˈrɪθmɪk ˈmɒdl/ | mô hình logarit |
| regression/rɪˈɡreʃn/ | hồi quy |
| parameters/pəˈræmɪtəz/ | tham số |
Growth that keeps slowing
- Some things improve fast at first, then crawl: a new skill, a filling reservoir, fading sound.
- The gains never stop, but each one is smaller than the last.
- That "fast then slow, forever" shape is exactly what a logarithm draws.
- So logarithmic models describe decelerating growth beautifully.
Building a logarithmic model
- A logarithmic model 对数模型 has the form $y = a\log x + c$.
- It shoots up steeply for small $x$, then rises ever more gently.
- Choose it when the data climbs quickly and then flattens without a ceiling.
- Two or three points can pin down the constants by hand.
A logarithmic model suits data that…
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Logs rise fast then flatten — perfect for growth that slows but never stops, like learning curves.
Regression finds the fit
- With scattered data, run a logarithmic regression 回归 on a calculator.
- It returns the best-fit $a$ and $c$ for the curve $y = a\log x + c$.
- The curve threads the trend without matching any point exactly.
- The result is a formula ready to interpret and use.
A curve that rises fast, then flattens
y = a·log(x − b) + c
A logarithmic model climbs steeply at first, then slows without ever stopping — ideal for data that keeps slowing down.
Technology can find a logarithmic ____ that best fits a data set.
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
A logarithmic regression returns the $a$ and $c$ in $y = a\log x + c$ that fit the points best.
Select all true statements about logarithmic models.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
A log rises without bound — it slows but has no ceiling. The other three are correct.
Interpret the parameters
- The parameters 参数 $a$ and $c$ carry real meaning.
- A larger $a$ stretches the curve, meaning a faster initial rise.
- The constant $c$ shifts the whole curve up or down to match the data's level.
- Reading these numbers turns the model back into a statement about the situation.
The parameters of a fitted model can be interpreted in terms of the real situation.
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Each constant carries meaning — a bigger $a$ means a faster initial rise, for example.
Which contrast is correct?
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They are inverses: the exponential accelerates, while its mirror the log decelerates.
Log versus exponential contexts
- An exponential fits things that accelerate: populations, compound interest, viral spread.
- A logarithm fits things that decelerate: sensory perception, diminishing returns.
- They are inverses, so their model shapes are opposite.
- Ask "is growth speeding up or slowing down?" to pick the right family.
A logarithmic model slows down but never stops — it has no upper limit. Do not use it for something that truly saturates at a hard ceiling (a different model handles that). It only fits growth that keeps rising, just more and more gently.
A study finds a test score rises with study hours as $S = 20\log_{10} h + 40$.
- At $h = 1$: $S = 20(0) + 40 = 40$.
- At $h = 10$: $S = 20(1) + 40 = 60$ — ten times the hours for $+20$ points.
- The gain from $10 \to 100$ hours is again just $+20$: clear diminishing returns.
A logarithmic model $y = a\log x + c$ fits data that rises quickly then keeps slowing. Find it with a logarithmic regression, interpret its parameters in context, and remember it has no ceiling — it decelerates but never stops rising.
Write from a log model value of 5
- y equals 2 times log base 10 of x, plus 1.
- Evaluate it at x equals 100.
For y equals 2 times log base 10 of x, plus 1, what is y when x equals 100?
4 plus 1 is 5.
What is y at x equals 100?
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
2 times 2 plus 1 is 5. Other options — 2 (the logarithm alone); 100 (the input).