Direct answer
A worked example can be done without live market data: pick a starting price, choose an ending price, and compare how far those points would be placed on a chart using either a linear scale or a logarithmic (log) scale. The main difference is what “equal distance” means: linear equal distances represent equal absolute price changes, while logarithmic equal distances represent equal percentage changes.
What the two scales mean (mechanics)
Linear scale: distance along the vertical axis is proportional to the absolute price (difference). If price goes from (P_0) to (P_1), the vertical movement is based on (P_1-P_0).
Logarithmic scale: distance is proportional to the log of price, often written as (\log(P)). If price goes from (P_0) to (P_1), the vertical movement is based on (\log(\tfrac{P_1}{P_0})). This makes equal percentage changes correspond to equal distances.
Important note: charting tools may use different log bases (commonly base-10 or natural log), but the comparisons remain consistent because log bases differ only by a constant factor.
Worked numerical example (with every assumption stated)
Assume:
- We are using the same chart height for both methods.
- The vertical position is compared using only the scale math (not any specific platform UI).
- Starting price is (P_0=100).
- We consider two scenarios that differ in absolute move but represent the same percentage move, and one scenario that has a different percentage move.
Scenario A: +10% from 100 to 110
- Ending price: (P_1=110).
- Linear distance (proportional): (P_1-P_0=110-100=10).
- Log distance (proportional): (\log(\tfrac{110}{100})=\log(1.1)).
Scenario B: another +10% from 200 to 220 (different absolute move)
Assume a new starting price (P_0=200) and ending (P_1=220). Again, this is +10%.
- Linear distance (proportional): (220-200=20).
- Log distance (proportional): (\log(\tfrac{220}{200})=\log(1.1)).
Comparison outcome:
- Linear: the distances differ (10 vs 20) because absolute changes differ.
- Log: the distances match because the percentage change is identical (both use (\log(1.1))).
Scenario C: +20% from 100 to 120
Assume (P_0=100), (P_1=120).
- Linear distance (proportional): (120-100=20).
- Log distance (proportional): (\log(\tfrac{120}{100})=\log(1.2)).
Comparison outcome:
- Linear: Scenario C has double the absolute change of Scenario A (20 vs 10), so it appears “twice as far” by this measure.
- Log: Scenario C does not scale linearly with absolute change; it depends on (\log(1.2)) versus (\log(1.1)).
Limitations and risks (what can go wrong)
- Axis configuration mismatch: Different chart settings can change how the scale is applied (for example, whether the axis is truly logarithmic or only the visual styling changes). Verification requires checking the chart’s axis mode.
- Comparisons depend on the same reference: The log distance uses (\tfrac{P_1}{P_0}). If you compare distances using different starting points without aligning references, conclusions may be misleading.
- No guarantee about interpretation quality: A log scale can make proportional growth look more uniform, but it can also hide absolute volatility changes because equal distances reflect equal percentage changes, not equal dollar (or pip) movement.
- Market and execution uncertainty: Real outcomes vary with spreads, execution quality, costs, and changing conditions. Historical visual relationships do not imply future results.
Verification and next question
To independently verify this concept, replicate the math with your own (P_0) and (P_1):
- Linear proportional movement: (\Delta_{lin}=P_1-P_0).
- Log proportional movement: (\Delta_{log}=\log(\tfrac{P_1}{P_0})).
If you want the next step, ask: when two chart tools show different “heights” for the same prices, which axis settings differ—and are both using a true logarithmic scale or just a transformed view?