4.3 The Y Axis

The x-axis decides when (or after how much activity) a glyph prints. The y-axis decides how vertical distance maps to price. CMT Level I asks you to define arithmetic versus logarithmic scaling, explain how arithmetic distorts long timeframes and wide ranges, show how scaling changes trendline slope, measured objectives, and visual risk/reward, choose a scale from timeframe, volatility, and asset characteristics, know when log is inappropriate or adds little, and calculate percentage and fold changes so additive steps are not confused with multiplicative ones.

This OpenExamPrep section is independent study material for those Level I y-axis topics.

Arithmetic versus logarithmic scaling

Arithmetic (linear) scaling: equal price differences occupy equal vertical space. The gap from 10 to 20 is drawn the same height as the gap from 110 to 120, because both are +10. The scale is additive.

Logarithmic scaling (semi-log chart: log price on Y, usually still time on X): equal price ratios occupy equal vertical space. The gap from 10 to 20 (a double, 2-fold, +100%) is drawn the same height as the gap from 80 to 160 (also a double), even though the second move is +80 points. The scale is multiplicative.

A fully log-log chart (log time and log price) is rare in classical TA. When a technician says "log chart" or "semi-log," they mean log price, arithmetic time.

How arithmetic distorts long timeframes and wide ranges

Distortion is not a software bug. It is what equal dollars do when the percent meaning of a dollar changes with the price level.

Suppose an equity is charted from 10 to 100 over a decade.

• 10 to 20 is +10 points and +100% (a double).
• 90 to 100 is +10 points and only +11.1%.

On an arithmetic long-term chart, those two +10 legs are the same height. The early doubling looks like a small wiggle near the bottom. The late 11% grind looks like a wall. Trendlines drawn across the whole history then describe dollar acceleration, not percent pace. Congestion at low prices is crushed; congestion at high prices is magnified.

The wider the multiple from low to high (2-fold, 5-fold, 10-fold), the worse the distortion. A two-week chart of a stock that moved from 50.00 to 51.50 (+3%) barely shows a difference between arithmetic and log — there is little multiplicative story to tell. A twenty-year chart of an index that went from 400 to 4,000 (10-fold) is a different picture on the two scales.

Volatility and asset characteristics enter the same way. A high-priced stock that routinely swings $15 in a day can look "wild" on arithmetic next to a $8 stock that swings $0.80, even if both moved about 10%. Log (or a percent chart) puts those swings on a comparable vertical.

Trendline slope, measured objectives, and visual risk/reward

Trendline slope. Two swing lows connected by a straight line are a constant dollar-per-time rise on arithmetic and a constant percent-per-time rise on log. The same two points can look steep on one scale and moderate on the other. A break of an arithmetic trendline is not automatically a break of the log trendline drawn through the same pivots, because the line is a different geometric object. Tools that assume straight-line channels therefore behave differently when you switch scales.

Measured price objectives. Classical measured moves often take a vertical height (a pole, a pattern) and project it from a breakout.

• On arithmetic, height is a dollar (point) distance. A 20-point flagpole projects 20 points above the breakout. From 40, the objective is 60. From 200, the objective is 220.
• On log, height is a ratio. A pole that doubled (2-fold) projects another 2-fold from the breakout. From 40, a full double objective is 80, not 60. From 200, a full double is 400, not 220.

If you measure in dollars on a log chart (or in percent on an arithmetic chart) without noticing, the target is a different number. Level I wants that distinction, not a catalog of every pattern's rule.

Visual risk and reward. Stops and targets are vertical distances on the chart you actually drew.

• A $4 stop under $20 is 20% risk. A $4 stop under $200 is 2% risk. Arithmetic draws those $4 distances as equal height, so the cheap stock's stop looks no larger than the expensive stock's stop. Log draws the 20% stop as a much larger visual drop than the 2% stop.
• Reward-to-risk that is 1:3 in dollars around one entry is also 1:3 in percent from that same entry (both are the same ratio of distances from one point). The scaling trap is comparing risk at different price levels, or reading a channel width that is constant in dollars as if it were constant in percent. A dollar-constant channel shrinks in percent as price rises; on log, a constant-percent channel stays even, while a constant-dollar band would narrow visually as price climbs.

In practice

You mark a long-term uptrend line under an equity that ran from 25 to 150. On arithmetic the line looks like it is steepening in the last two years because each year's dollar gain is larger. On log the same years may look like a steady percent climb or even a flattening if percent gains slowed. Your measured objective from a 25-point continuation pattern is +25 points on arithmetic and a percent/fold projection on log. The stop you sketched as "about this much space under the line" is a dollar gap on one chart and a percent gap on the other. Same pivots, different geometry.

When to use each scale

Choose from timeframe, volatility, and asset — not from a superstition that "professionals always use log."

Timeframe: the longer the history and the larger the fold change from low to high, the more log earns its keep. Volatility: if typical swings are large in percent, log (or percent) keeps them comparable across years. Asset: growth equities and long index histories often need log; a short-rate or a spread that can sit near zero often does not.

When logarithmic scaling is inappropriate or adds little

Adds little. If the window's entire percent range is tiny — a stock from 100 to 103 in a week, a bond future that traveled a handful of ticks — the log transform is almost linear over that slice. You will not see a new trendline story. Switching to log to look "more professional" is wasted motion.

Inappropriate.

• Non-positive prices. The logarithm is not defined for zero or negative values in the real numbers used on these charts. Spreads, some difference series, and instruments that can trade through zero belong on arithmetic (or a signed scale). Do not force log.
• Already-ratio series, used carelessly. A relative-strength ratio is already multiplicative. You may log a ratio (that is a further transform), but logging it without noticing you doubled the multiplicative lens is a construction error, not a default.
• Dollar geometry required. If the question is "how many points to the strike / to the stop / to the target in ticks," arithmetic matches the contract. Log would relabel the same ticks as changing percents.

Percentage change versus fold change

Percentage change (simple) is additive in return units only after you compute it:

% change = (New − Old) / Old × 100

Fold change (multiple) is multiplicative:

Fold = New / Old

A 2-fold move is a double (+100%). A 1.5-fold move is +50%. A 3-fold move is +200%, not "three times +100% stacked as three equal arithmetic steps."

Required numeric example

Start at 50, then 100, then 150:

On arithmetic, 50 → 100 and 100 → 150 are equal heights (+50 and +50). On log, they are not equal. The first leg is a double; the second is only 1.5-fold.

Equal log height to the first leg would be another double: 100 → 200, not 100 → 150.

Check with common logarithms (any log base preserves the equality of ratios):

• log10(100) − log10(50) = 2.0000 − 1.6990 = 0.3010
• log10(150) − log10(100) = 2.1761 − 2.0000 = 0.1761 (smaller)
• log10(200) − log10(100) = 2.3010 − 2.0000 = 0.3010 (matches the first doubling)

50 → 100 is therefore +100% / 2-fold. 100 → 150 is +50% / 1.5-fold, not another equal arithmetic step on a log chart.

A second trap: two successive +50% legs. 100 × 1.5 = 150, then 150 × 1.5 = 225. The second +50% is +75 points, not another +50 points. Percentages compound; they do not add like dollars unless you convert to log returns.

Exam traps

• Treating +50 points as a constant percent. It is +100% from 50 and +50% from 100.
• Projecting a dollar measured move on a log chart (or the reverse) without converting.
• Forcing log on a spread that can be negative.
• Assuming a short, quiet chart "needs" log. If the percent range is tiny, log adds little.

The next section adds volume and open interest as plotted series — how they are drawn, not yet how they confirm a trend. Independent CMT Level I practice by OpenExamPrep is at /practice/cmt.