Semi-Log Graph Paper

One axis stretched, one axis not

The horizontal axis is ordinary: equal steps, equal distances. The vertical one is logarithmic, so each decade — 1 to 10, 10 to 100, 100 to 1000 — takes the same height on the page.

Inside each decade the minor lines sit at 2, 3, 4 and so on up to 9, which is why they crowd together as they approach the top of every block. That crowding is the log scale showing its working.

What it is for

Data that grows by multiplication rather than by addition.

Bacterial cultures, compound interest, radioactive decay, viral spread, Moore's law, the loudness of anything. On linear paper such a series either flattens into the axis at one end or shoots off the top at the other, and usually both. Here it fits, and more usefully it straightens: an exponential relationship plots as a straight line on semi-log paper, and the slope of that line tells you the growth rate.

That is the whole trick. A curve is hard to judge by eye and a straight line is not, so a scientist checking whether a process really is exponential reaches for this sheet before reaching for a calculator.

Choosing your decades

Count the orders of magnitude your data spans, then add one for headroom. Readings running from 5 up to 3000 cross three decades, so four decades leaves you room at both ends of the sheet.

Too few and the data runs off the sheet. Too many and it huddles into a single band near the middle, wasting most of the paper you just printed. Four is the default because it suits most school and lab work.

The linear divisions

Ten by default, which reads naturally as ten days, ten weeks, ten trials. Anything from 2 to 20 works. Choose the number that matches your sampling interval rather than one that looks tidy — a chart whose gridlines land on the actual measurement points is far quicker to plot on and much harder to misread.

If both axes need it

Use log-log paper instead, where both scales are logarithmic. That is the sheet for power laws, where y is proportional to x raised to some power — those plot straight on log-log and curved on semi-log.

Rule of thumb: exponential, semi-log. Power law, log-log.

Print at 100% on A4, US Letter or US Legal so the decades keep their true height.