Axis crimes

The logarithmic unit

The axis is linear. The unit is already a logarithm.

a.k.a. pH · decibels · the Richter scale · moment magnitude · stellar magnitude · f-stops · semitones · the log in the unit · the already-logged variable · ratio units

A log axis at least announces itself: the ticks read 1, 10, 100, and a reader who looks can see what was done. This one hides a step earlier, inside the measurement, where no axis setting can reach it. pH, the decibel, earthquake magnitude, stellar magnitude, the f-stop and the semitone are not quantities but logarithms of quantities, agreed on long ago precisely because the numbers underneath span too many orders of magnitude to carry around. Plot one on a perfectly ordinary linear axis and that axis is quietly doing what a log axis does: equal distances are equal ratios, so every gap the eye measures is a multiplication and no difference on the chart is an amount. Three things follow, and each of them survives an audit. Adding logs multiplies the quantities, so a mean of pH readings is the geometric mean of the acid rather than the arithmetic one, and a gap quoted in “points” is a ratio rather than an amount. A curve that flattens has not slowed down — it has stopped multiplying as fast, while the quantity underneath can be accelerating, and the flat stretch is often where most of it arrives. And several of these units carry a minus sign as well, so the line falls while the thing it measures rises. Unlike a truncation, none of it can be fixed by widening the range: draw the pH from zero, or across the whole 0–14, and the picture is flatter rather than fairer, because the compression lives in the unit and not in the frame.

How to spot it

Ask what the unit is a logarithm of. pH is minus the log of hydrogen-ion activity; a decibel is ten times the log of a power ratio; a magnitude step is ×31.6 in earthquake energy and ×2.512 in starlight; an f-stop is ×2 in light and a semitone ×1.0595 in frequency.
Evenly spaced, innocuous ticks — 6.0, 5.5, 5.0 — with no 10, 100 or 1,000 anywhere on the axis. That is the whole disguise, and it is why a log axis is far easier to catch than a logged unit.
A curve that goes flat. In a logged unit, flat means the ratio stopped moving quickly; the amount per week can be climbing throughout, and the flat stretch is often where most of it arrives.
Arithmetic on the values: an average pH, a mean magnitude, a difference quoted in points or dB, a percentage change in either. Adding logs multiplies the quantities, so every one of those is answering a multiplicative question — convert, do the sum, convert back.
Bars. These units have no meaningful zero to grow from (pH 0 is one mole per litre and pH runs negative; 0 dB is a chosen reference), so a bar length encodes a ratio to whatever baseline was picked, never an amount.
A minus sign in the definition. pH and stellar magnitude both run backwards, so the smaller number is the larger quantity and the chart falls as the thing rises — an inversion nobody had to draw.
Look for a second measurement of the same thing in a linear unit sitting in the same source: titratable acidity beside pH, sound intensity beside decibels, seismic moment beside magnitude. Where the two disagree about the shape of the curve, the unit is doing it.
The tell that separates it from a truncated axis: widen the axis to the full range of the unit and nothing about the shape changes.

The fix

Plot the quantity the reader is being asked about. If the question is how much acid, how much sound energy, how much light, put the un-logged quantity on a linear zero-based axis and let it be the hockey stick it is — the change of shape between the two pictures is not a distortion being introduced, it is the one the logged unit was applying all along. Where the range genuinely spans orders of magnitude and the linear chart collapses into a line along the bottom, the honest answer is a log axis drawn by the rules under “The log scale” — which at least prints its decades on the ticks, where a logged unit prints nothing at all. Keep the logged unit where it is the thing people actually act on, because often it is: pH 4.6 is where the rule for acidified foods is drawn, an 85 dB exposure limit is a real limit, and a magnitude is what a building code is written against. The distinction worth holding is between a threshold and an amount — a logged unit answers which side of the line, and only the un-logged quantity answers how far — and even a threshold is worth reading with its linear neighbour in view, since microbial inhibition tracks undissociated acid as well as pH. So say in the caption what the unit is a logarithm of and what one step of it multiplies, the way you would print any other unit, and where a decision turns on an amount, show both charts and take the number off the linear one. Never let the logged values through a mean, a sum, a difference or a growth rate on the way to a slide.

In the gallery

$Dark navy briefing slide, masthead “Process control briefing · spontaneous sauerkraut fermentation · 15 °C · 42 days” on the left and “pH · 13 sampling points · measured in triplicate” on the right, headlined “The kraut was made in week one.” A standfirst says there are thirteen sampling points, one line and every reading on the page; that the figures are the laboratory’s own, from a spontaneous sauerkraut fermentation held at 15 °C for 42 days with pH measured in triplicate at each point; that nothing is smoothed, interpolated, indexed or rebased, no point dropped and none averaged into a bucket; that the days are plotted where they fall so a week of calendar occupies a week of chart, and the axis is linear, evenly ticked and unbroken. It then reads off the chart: pH falls from 5.87 to 3.93 in the first seven days — 1.94 of the 2.51-point drop, 77.3% of it — and then goes flat, moving 0.57 of a point over the remaining five weeks, of which four — days 14 to 42 — are worth 0.38 of a point between them. The main panel, headed “The fall in pH, day 0 to day 42”, subtitled “All 13 sampling points · linear time axis, days plotted where they fall · linear pH axis, 3.0 to 6.0 in even half-point steps · each point the mean of three replicates”, is a gold line chart with a marked point at each of the thirteen samples: 5.87 at day 0, 5.76, 5.71 and 5.70 through day 2, then a steep fall through 5.15, 4.69 and 4.36 to 3.93 at day 7, after which the line runs almost horizontally through 3.74, 3.67, 3.54 and 3.49 to end at 3.36 on day 42. A dashed vertical marks day 7. A gold dashed rule at 5.87 runs from day 0 to day 7 under the label “Week one: −1.94 points · 77.3% of the whole drop”, and a pale dashed rule at 3.93 runs from day 7 to day 42 under “Days 7–42: −0.57 of a point”. Six cards run down the right. “The drop, in one figure” prints 1.94 in large type over a note that this is the points of pH lost in the first seven days out of a 2.51-point fall across the whole 42 — 77.3% of the total — and that the remaining five weeks are worth 0.57 of a point between them. “Read-out for the production committee” says the curve is essentially flat from day 7, that every week after it buys a fraction of what the first week bought while keeping a tank, a headspace and a cold room out of service, and that days 14 to 42 are worth 0.38 of a point or 15.1% of the drop — recommending the standard cycle be cut from 42 days to 14 and the capacity re-used. “Schedule options, in the order the chart gives them” tabulates six pull points with their pH and cumulative share of the drop: day 7 at 3.93 and 77.3%, day 14 at 3.74 and 84.9%, day 21 at 3.67 and 87.6%, day 28 at 3.54 and 92.8%, day 35 at 3.49 and 94.8%, day 42 at 3.36 and 100.0%. “Why the axis starts at 3.0” explains that pH has no zero to start from — pH 0 is one mole of hydrogen ion per litre and the scale runs negative below it — so the axis is drawn across the range the readings occupy, in even half-point steps, with no break and no cap. “The 13 sampling points, as measured” lists every reading in two columns. “How to read the chart” says to follow the line left to right, that steeper means the reading moved faster, and that the gap between two points read off the axis is the change in pH between them. A final card gives the basis of measurement: Tlais, Lemos Junior, Filannino, Campanaro, Gobbetti and Di Cagno, Microbiology Spectrum 10(4):e00168-22 (2022), Table 1, with pH values the means of three replicates. A method line runs along the foot: bench pH meter, three replicates per sampling point, means plotted; 13 points across 42 days; linear time axis; linear pH axis in even half-point steps; no smoothing, no interpolation, no exclusions. A footnote credits the source, notes all 13 pH readings are as published, and says the slide is a demonstration drawn by us in the manner of a process control briefing — the committee, the read-out and the schedule options are invented, and every figure behind them is real.$

The 77% of the pH drop that is 31% of the acid

The logarithmic unit