One step, one multiplication
On an ordinary scale, each step adds the same amount: 6 metres is one metre more than 5. On a logarithmic scale, each step multiplies by the same factor: each whole number on the pH scale is ten times the previous one. The scale number is the exponent, the power of ten, rather than the quantity itself. The logarithm calculator converts in both directions, and the scientific notation converter shows the powers of ten that sit underneath.
Difference of n steps on a base-10 scale = a factor of 10ⁿ. Two steps = 100 times; three steps = 1,000 times.
Earthquake magnitude
The USGS explains that each whole-number increase in magnitude is a tenfold increase in the amplitude measured on a seismogram and corresponds to about 32 times more energy released:
| Magnitude difference | Ground motion amplitude | Energy released |
|---|---|---|
| 1 | 10 × | about 32 × |
| 2 | 100 × | about 1,000 × |
| 3 | 1,000 × | about 33,000 × |
The USGS also separates magnitude from intensity: an earthquake has a single magnitude, while intensity measures the shaking at each place on the Modified Mercalli scale and varies with distance. And the familiar name is out of date: the USGS says the Richter scale is not commonly used anymore except for small local earthquakes, and that moment magnitude is the more accurate measure.
pH
The pH scale runs from 0 to 14, with 7 neutral, values below 7 acidic and values above 7 basic. The USGS: each number represents a tenfold change in acidity or basicness, so water at pH 5 is ten times more acidic than water at pH 6.
| pH difference | Factor in acidity or basicness |
|---|---|
| 1 | 10 × |
| 2 | 100 × |
| 3 | 1,000 × |
The pool chlorine and pH guide shows why a pool that drifts from 7.4 to 8.4 has changed far more than the decimal suggests.
Decibels
Sound uses a base-10 scale in tenths: NIOSH notes that a sound 10 dB louder than another is ten times more intense. Smaller steps follow from the same rule:
| Decibel difference | Intensity factor |
|---|---|
| 3 dB | about 2 × |
| 10 dB | about 10 × |
| 20 dB | about 100 × |
| 30 dB | about 1,000 × |
That 3 dB, about double, is the step NIOSH uses to halve the safe listening time; the how loud is too loud guide has the time table.
Reading logarithmic numbers correctly
- Subtract, then exponentiate. The gap between two values tells you the factor: magnitude 8 against 6 is two steps, 100 times the ground motion.
- Never average the scale numbers to average the quantities; convert to the underlying values first.
- Small changes are big. A tenth of a step on a base-10 scale is still about 26 percent.
- Know the base. Earthquake amplitude, pH and decibel intensity all use powers of ten, but the energy factor for earthquakes is about 32 per step, so say which quantity you are comparing.
Frequently asked questions
What is a logarithmic scale?
A scale on which each step multiplies the quantity by a fixed factor instead of adding a fixed amount. On the pH scale each whole number is a tenfold change; on the earthquake magnitude scale each whole number is ten times the ground motion. Equal distances on the scale mean equal ratios, not equal differences.
How much stronger is a magnitude 7 earthquake than a 6?
The USGS says each whole-number step in magnitude is a tenfold increase in the amplitude measured on a seismogram and about 32 times more energy released. So a magnitude 7 releases about 32 times the energy of a 6, and about 1,000 times that of a 5.
Is pH 4 twice as acidic as pH 5?
No, ten times. The USGS explains that each number on the pH scale represents a 10-fold change in acidity or basicness, so water at pH 5 is ten times more acidic than water at pH 6, and pH 4 is a hundred times more acidic than pH 6.
Why are decibels logarithmic?
Because the ear handles an enormous range of sound intensities. NIOSH notes that a sound 10 dB louder is ten times more intense; a logarithmic scale compresses that range into manageable numbers, and 3 dB more is close to double the intensity.
Why use logarithmic scales at all?
They fit quantities that span many powers of ten into a short range of numbers, and they turn multiplication into addition: adding 1 to a magnitude or a pH multiplies the underlying quantity by the same factor every time, wherever on the scale you start.
Sources
- USGS Earthquake Hazards Program: Earthquake magnitude, energy release and shaking intensity (each whole-number step is a tenfold increase in measured amplitude and about 32 times more energy; magnitude versus intensity; the Richter scale replaced by moment magnitude)
- USGS Water Science School: pH and water (scale from 0 to 14, 7 neutral; each number represents a 10-fold change; pH 5 is ten times more acidic than pH 6)
- CDC NIOSH: About noise and hearing loss prevention (a sound 10 dB louder than another is ten times more intense)