The inverse of an exponential
The logarithmic function f(x) = log_b x is the inverse of g(x) = bˣ. Their graphs are reflections in the line y = x.
| y = 2ˣ | y = log₂ x | |
|---|---|---|
| (0, 1) | ↔ | (1, 0) |
| (1, 2) | ↔ | (2, 1) |
| (2, 4) | ↔ | (4, 2) |
| (3, 8) | ↔ | (8, 3) |
| (−1, ½) | ↔ | (½, −1) |
Swap the coordinates to get the inverse.
Features of y = log_b x (b > 1)
| Feature | Value |
|---|---|
| Domain | x > 0 |
| Range | all real numbers |
| x-intercept | (1, 0) |
| Vertical asymptote | x = 0 |
| Behaviour | increasing, but more and more slowly |
For 0 < b < 1, the graph decreases instead.
Slow growth
log₁₀ x: when x goes from 10 to 100 to 1,000 to 1,000,000, y only goes 1, 2, 3, 6. Logarithms grow very slowly, which makes them useful for very wide ranges.
Transformations
y = a log_b(x − h) + k
- Vertical asymptote moves to x = h.
- Domain: x > h.
Example: y = log₂(x + 3) − 1
- Asymptote x = −3; domain x > −3
- Key point (1, 0) moves to (−2, −1)
Finding the domain
Set the argument > 0.
- y = log(5 − x) → x < 5
- y = ln(2x + 8) → x > −4
Logarithmic scales
A logarithmic scale compresses huge ranges.
| Scale | Measures |
|---|---|
| Richter/moment magnitude | earthquakes |
| Decibels (dB) | loudness |
| pH | acidity |
| Stellar magnitude | brightness of stars |
Example: pH = −log[H⁺]. Lemon juice (pH 2) has 10⁵ = 100,000 times the hydrogen ion concentration of pure water (pH 7).
Example: A 60 dB conversation vs a 90 dB lawn mower: 30 dB difference → 10³ = 1,000 times the sound intensity.
Worked examples
Example 1: Domain of y = log(x − 4)? x > 4.
Example 2: Vertical asymptote of y = log₃(x + 6) + 2? x = −6.
Example 3: An earthquake of magnitude 7 vs 5: ground motion ratio? 10² = 100 times.
Common mistakes
- Allowing x ≤ 0 in the domain.
- Confusing the vertical asymptote of logs with the horizontal asymptote of exponentials.
- Thinking a logarithmic scale is linear.
Try it yourself
- Inverse of y = 5ˣ?
- x-intercept of y = log₄ x?
- Domain of y = log₂(3 − x)?
- Asymptote of y = ln(x − 1)?
- A sound goes from 50 dB to 70 dB. How many times more intense?
Answers: 1) y = log₅ x 2) (1, 0) 3) x < 3 4) x = 1 5) 100 times
Quick recap
- A logarithmic function is the inverse of an exponential.
- Domain x > 0; vertical asymptote x = 0; x-intercept (1, 0).
- Logarithmic scales handle huge ranges.