Understanding linear and logarithmic scales
Place the same numbers on linear and logarithmic rulers and test how differences and ratios change their screen positions.
Some rulers add; this ruler counts tenfold steps
On a usual ruler, equally spaced marks might be 1, 2, 3, 4. On a logarithmic ruler, they can be 1, 10, 100, 1,000. Each gap then means ten times as much, not ten more.
Work it out, one step at a time
Count the multiplications
1 × 10 = 10; 10 × 10 = 100; 100 × 10 = 1,000
There are three equal-ratio steps from 1 to 1,000.
Find the position
10: 1 / 3; 100: 2 / 3
Ten is one third of the way along this log ruler; one hundred is two thirds.
Words made simple
- Ratio
- How many times one quantity is another, found by division.
Think about it: Is 100 halfway between 10 and 1,000 on a log ruler?
Yes. 100 ÷ 10 = 10 and 1,000 ÷ 100 = 10. Both sides are one tenfold step.
Where the numbers come from: NIST. The illustration and worked model are made for this lesson; assumptions are stated above.
A linear ruler adds; a logarithmic ruler multiplies
A linear ruler gives equal space to equal changes: 1, 2, 3, 4. A logarithmic ruler gives equal space to equal ratios: 1, 10, 100, 1,000. Both keep the same numbers. They change how those numbers occupy the screen.
Predict where 10 will sit between 1 and 1,000. On a linear ruler it is only 9/999 of the way across—less than 1%. On a base-10 logarithmic ruler it is one third of the way across, because the full interval contains three tenfold steps.
Try it: same values, different ruler
Does equal spacing mean an equal difference?
On this 1–1,000 ruler, the small values crowd at the left. Their labels are listed separately below so they remain readable.
- 1
- 10
- 100
- 1,000
The calculation behind the picture
plinear = (x − min) / (max − min)
plog = log₁₀(x / min) / log₁₀(max / min)
Here min = 1 and max = 1,000. Position p runs from 0 at the left to 1 at the right. For x = 100, the logarithmic calculation is log₁₀(100)/log₁₀(1,000) = 2/3. Ratios require positive lengths; zero has no finite position on this logarithmic ruler.
Why a cell and a planet can share one journey
A DNA diameter of about 2 nm and Earth’s diameter of 12,742 km differ by roughly 6.37 × 10¹⁵ in length. A literal linear screen would make the DNA invisible. ScaleTrip uses logarithmic spacing and separately compresses artwork size. A similarly sized picture therefore means “recognizable illustration”, while the labels and ratios supply the measurement.
Check your understanding
Is 100 halfway between 10 and 1,000? On a logarithmic ruler, yes: each side is a factor of ten. On a linear ruler, the midpoint is 505. Equal visual gaps tell you about the chosen ruler, not a universal physical spacing.
NIST — Metric (SI) prefixes · NHGRI — DNA Day transcript, question 296 · NASA JPL — Planetary Physical Parameters
The positions, diagrams and worked examples on this page are calculated and authored for ScaleTrip.
Explore the outward journey with this ruler · Next learning questionCan one metre and cosmic distances fit on one screen?
Keep the numbers fixed and change how the ruler allocates space. On a 1 m to 10²⁷ m ruler, a light-year sits about 59.2% across the logarithmic view, yet below one hundred-billionth of the linear width.
| Length definition | Value | Position from left |
|---|---|---|
| 1 m | 1 m | |
| 1 km | 1 km | |
| Mean Earth–Moon separation | 3.844e+8 m | |
| 1 au | 1.496e+11 m | |
| 1 light-year (Julian year) | 9.461e+15 m | |
| Chosen ruler endpoint | 1.000e+27 m |
What the algorithm changes
plinear = (x − min) / (max − min)
plog = log₁₀(x / min) / log₁₀(max / min)
This experiment reuses ScaleTrip’s scalePosition function. It computes a position from 0 to 1, then maps it to a 680-unit drawing window. Changing the upper endpoint moves values beyond it out of view; it does not clamp them onto a misleading boundary.
One light-year here equals 299,792,458 m/s × 86,400 s/day × 365.25 days. The 10²⁷ m upper endpoint is a chosen ruler boundary near cosmic orders of magnitude, not a measured diameter of the universe. Neither this ruler nor the outward journey is a map or an image of every distance at one linear scale.
Compare 1 km in the two modes, first with a 10³ m endpoint and then 10²⁷ m. The value stays 1,000 m. Its share of the screen changes. Can a large-looking gap tell you an absolute distance without knowing the ruler?
- NASA Space Place — Moon distance — Recorded source check:
- NIST — Definitions of SI base units — Recorded source check:
- NASA JPL — Astrodynamic Parameters (au, Julian year, day) —