Logarithmic Functions
| English | Chinese | Pinyin |
|---|---|---|
| logarithmic function | 对数函数 | duì shù hán shù |
| base | 底数 | dǐ shù |
| domain | 定义域 | dìng yì yù |
| vertical asymptote | 竖直渐近线 | shù zhí jiàn jìn xiàn |
| end behavior | 末端行为 | mò duān xíng wéi |
The slow-and-steady curve
- The exponential rockets upward; its mirror, the logarithm, does the opposite.
- A logarithm rises forever but slower and slower, never flattening to a stop.
- It clings to the y-axis on the left and creeps upward on the right.
- This gentle shape models sound loudness, acidity, and the "feel" of big numbers.
The graph of a logarithm
- A logarithmic function 对数函数 is $f(x) = \log_b x$, the inverse of $b^x$.
- Its graph passes through $(1, 0)$, since $\log_b 1 = 0$ for every base.
- To the right of $1$ it is positive; between $0$ and $1$ it is negative.
- Read the shape as the exponential's reflection across $y = x$.
Shape the logarithmic curve
y = a·log(x − b) + c
This log curve climbs slowly and clings to the y-axis. It exists only for x greater than its vertical asymptote.
Domain and vertical asymptote
- The domain 定义域 is $x > 0$ — you cannot take the log of zero or a negative.
- As $x \to 0^+$, the output plunges to $-\infty$: a vertical asymptote 竖直渐近线 at $x = 0$.
- The graph never touches the y-axis, only hugs it more and more tightly.
- Its range, by contrast, is all real numbers.
The domain of $f(x) = \log_b x$ is…
You can only take the log of a positive number, so the domain is $x > 0$.
A logarithmic function has a ____ asymptote at $x = 0$.
As $x \to 0^+$ the output dives to $-\infty$, so $x = 0$ is a vertical asymptote.
The graph of $y = \log_b x$ crosses the x-axis at…
Since $b^0 = 1$, we have $\log_b 1 = 0$: the x-intercept is always at $x = 1$.
Select all true statements about $y = \log_b x$ (with $b > 1$).
It rises without bound, so it has no maximum. The other three are correct.
The base sets the direction
- If the base 底数 $b > 1$, the log function is increasing (slowly rising).
- If $0 < b < 1$, it is decreasing instead.
- A bigger base makes the curve climb even more gently.
- Whatever the base, the curve still passes through $(1, 0)$ and has the same asymptote.
A logarithmic function keeps increasing forever, but more and more slowly.
It rises without bound as $x \to \infty$, yet its slope flattens — the opposite of exponential speed-up.
End behavior: unbounded but slow
- As $x \to \infty$, the end behavior 末端行为 is $\log_b x \to \infty$ — it never levels off.
- But it grows so slowly that reaching $y = 10$ can take an enormous $x$.
- Compared with an exponential's explosion, the logarithm barely moves.
- That is exactly why logs are used to compress huge ranges of numbers.
A logarithm's slow growth is not the same as "having a ceiling". It keeps rising forever — there is no horizontal asymptote and no maximum. It just takes longer and longer to climb each additional unit.
Describe $y = \log_2 x$ at a few points.
- $\log_2 1 = 0$ (the x-intercept), $\log_2 2 = 1$, $\log_2 4 = 2$, $\log_2 8 = 3$.
- To gain each $+1$ in output, the input must double — growth is slowing.
- Near $x = 0$ the curve dives down the vertical asymptote toward $-\infty$.
A logarithmic function $\log_b x$ has domain $x > 0$, a vertical asymptote at $x = 0$, and passes through $(1, 0)$. The base decides increasing vs decreasing, and its end behavior is unbounded but ever-slowing growth — the mirror of an exponential.