Why do long histories mislead on a linear scale?

This is the classic beginner trap, and it catches people for years without them noticing. Every chart has a price axis, and that axis can be drawn two ways. On short histories the difference is invisible. On long ones it changes the shape of everything.

The two scales

Linear (arithmetic) scale. Equal dollar distances take equal vertical space. The gap from $10 to $20 looks exactly as tall as the gap from $110 to $120.

Logarithmic (log) scale. Equal percentage distances take equal vertical space. The gap from $10 to $20 (a double) looks exactly as tall as the gap from $100 to $200 (also a double).

Investors and traders care about percentages — a 10% move means the same thing to your account whether the stock is $10 or $1,000. A linear axis systematically disagrees with that.

A worked example

Take an illustrative stock over twenty years: $2 → $20 → $200.

Both legs are a tenfold gain. Owning it through the first leg and through the second produced exactly the same percentage return.

• On a linear chart, the first leg ($2 to $20, an $18 move) is a barely visible crawl along the bottom. The second leg ($20 to $200, a $180 move) fills the entire screen and looks like a vertical explosion. The visual story: nothing happened for a decade, then it went parabolic.
• On a log chart, the two legs are the same height, and the line looks like a steady climb. The visual story: it compounded at a fairly consistent rate for twenty years.

The second story is the accurate one. The first is an artefact of the axis. And notice how easily the linear picture invites words like "parabolic" or "unsustainable" — words about the drawing, not about the company.

Here is that path drawn both ways. The two schematics carry the same twenty-five closes — the same $2, the same $20, the same $200 — and the only thing that changes between them is the price axis.

Put a finger on the screen and compare. On the linear axis the first decade is a flat crawl along the bottom and the second is a cliff; on the log axis the two legs are the same height, because they are the same gain. Neither picture is lying and neither was drawn from a kinder set of numbers. The first one is answering a question about dollars while you were asking one about percentages.

Where this bites hardest

• Long-term index charts. Nearly every viral "the market is in a bubble" chart is a decades-long index on a linear axis. Redraw it on log and the terrifying vertical wall usually becomes an ordinary slope.
• Old crashes. On a linear multi-decade chart, a 40% crash in the 1980s is a hairline scratch while a 15% dip last year is a canyon. On log, both are drawn at their true relative severity.
• Trendlines. A trendline drawn on a linear chart and the same trendline drawn on log are different lines touching different candles. Traders who use trendlines need to fix one scale and stay on it — otherwise their own tool changes underneath them.

The workable rule

Short horizon or small percentage range → the two scales are nearly identical, use whichever. Anything spanning years, or any instrument that has multiplied several times over → log. Say the scale out loud before reading a long chart, the same way you check the axis units on any scientific plot.

Try it now

1. Measure the two legs of the linear schematic above against your screen and write down the ratio of the two drawn heights — that number is the size of the distortion, on a picture that is telling you the exact truth. Measure the same two legs on the log one: the ratio should come out at 1.
2. Now the real thing, on a linear axis like the first schematic: the longest history we hold for a stock that has multiplied many times over. Find the decade that looks like nothing happened, then Measure it end to end. The percentage the tool reports is what that flat-looking stretch actually returned, and it will not match the picture.

3. Do the same measurement on a broad index over the same window. If it has not multiplied much, the picture and the arithmetic will agree — that null result is worth seeing too, because it shows exactly when the trap does and does not apply.