Log Graph Paper
Semi-log and log-log graph paper with 1 to 6 cycles on each logarithmic axis, heavier decade lines and finer subdivisions only where they stay readable. Choose which axis is logarithmic, the square size of the linear axis and the page, then print it true to scale.
Preview and download
Scaled-down preview; the printout is full size. Dashed lines show the margins and don’t print.
Print at 100% or Actual size, not “Fit to page”. Then check the ruler at the foot of the sheet: it should measure exactly 4 in.
- Sheet
- Letter, 8.5 × 11 in, portrait
- Pattern
- 3 cycles × 70 divisions
- Ruler
- 4 in, in the bottom margin
- 1 page, vector, no watermark
Paper
Scales
Lines and colour
Margins
From each paper edge to the pattern. Most printers can’t print the outer 0.2 in or so of a sheet.
Header and mixed pages
Pages and printing
How to make log paper
- Semi-log or log-log. Under Paper type, semi-log has one logarithmic axis and one ordinary one; log-log has two. For semi-log, choose whether the log scale runs up the page (y) or across it (x).
- Cycles. One cycle per power of ten your data spans, from 1 to 6 on each log axis.
- The linear axis. On semi-log paper, set its square size and a heavier line every few squares. The starting sheet, with 3 cycles and 1/10 in squares with a heavier line every inch, comes out as 3 cycles × 70 divisions on Letter.
- Page and print. Portrait gives a tall log axis, landscape a wide one. Print at 100%, especially if you will measure slopes with a ruler.
How a log axis works
On a log axis, equal distances stand for equal ratios rather than equal differences: 1 to 10 takes the same space as 10 to 100. Within each decade, the lines for 2, 3, 4 … 9 crowd together towards the end of the decade, because each sits at log₁₀ of its value along the decade (2 is 30.1% of the way up, 5 is 69.9%). The generator draws every whole number, adds finer lines at steps of 0.1, 0.2 or 0.5 wherever they stay at least about 1 mm apart, and numbers each decade 1 to 9, leaving out some numbers on short decades where they would touch. How long a decade is depends on the number of cycles and the page:
| Cycles | Range covered | Letter, up the page | Letter, across | A4, up the page |
|---|---|---|---|---|
| 1 | 1 to 10 | 9.94 in | 7.45 in | 269 mm |
| 2 | 1 to 100 | 4.97 in | 3.73 in | 135 mm |
| 3 | 1 to 1,000 | 3.31 in | 2.48 in | 90 mm |
| 4 | 1 to 10,000 | 2.49 in | 1.86 in | 67 mm |
| 5 | 1 to 100,000 | 1.99 in | 1.49 in | 54 mm |
| 6 | 1 to 1,000,000 | 1.66 in | 1.24 in | 45 mm |
Portrait sheets with the starting margins (1/2 in on Letter, 10 mm on A4), measured from the decade lines the generator draws. Any starting power of ten works: 3 cycles can just as well run from 0.01 to 10.
Semi-log paper: growth, decay and frequency
Anything that grows or shrinks by a constant factor per unit of time plots as a straight line on semi-log paper, which makes the rate easy to read and departures from it easy to see:
- Bacterial growth curves. Cell counts against time: the exponential (log) phase is the straight part, and the lag and stationary phases show as bends.
- Decay and half-lives. Radioactive decay, drug concentrations in first-order elimination, and cooling towards room temperature (plotting the temperature difference) are straight lines falling with time.
- Money and populations. Compound growth at a steady rate is a straight line, so a change in slope is a change in the growth rate.
- Frequency response. Bode plots and audio response curves put frequency on the log axis and gain in decibels on the linear one. Decibels, like pH, are already logarithmic units, so they belong on the linear axis; pH against the concentration of a dilution series is another semi-log plot.
- Grain size. Sieve analyses of soil and aggregate plot the percentage passing against sieve size on a log axis.
Reading a doubling time. A factor of 2 is log₁₀ 2 = 0.301 of a decade. On the starting Letter sheet a decade is 3.31 in tall, so a doubling is a rise of about 1 in: find two points on the line that far apart vertically and read the time between them. A halving is the same distance downwards.
Log-log paper: power laws
If y = a·xb, then log y = log a + b·log x, so the data falls on a straight line of slope b. The area of a circle against its diameter has slope 2; a pendulum's period against its length has slope 1/2; a planet's orbital period against the size of its orbit has slope 3/2. A straight line on log-log paper is the quickest test that a relationship is a power law, and its slope gives the power.
To read the slope with a ruler, both axes need decades of the same length. The default 2 × 3-cycle Letter sheet has decades of 3.69 in across and 3.28 in up, so slopes measured with a ruler would read 11% too low. With 3 cycles across and 4 up, both are 2.46 in. Otherwise, read two points' values and calculate the slope from their logarithms.
Printing it
Values are read off the printed scale, so a small scaling error doesn't change them, but it does change slopes and distances you measure with a ruler, and printers can scale the two directions slightly differently. Print at Actual size and check the ruler at the foot of the sheet; the printing guide covers the settings for each browser and PDF reader.
Frequently asked questions
What does “cycles” mean on log graph paper?
A cycle is one decade: the stretch of axis in which values grow tenfold, from 1 to 10, 10 to 100, and so on. Every cycle is the same length on the page, so 3-cycle paper spans a factor of 1,000 and 6-cycle paper a factor of a million. More cycles cover a wider range, but each one is shorter and harder to read precisely.
How many cycles do I need?
Count the powers of ten your data crosses. Values from 3 to 4,000 fall in 1–10, 10–100, 100–1,000 and 1,000–10,000, so they need 4 cycles. Round the smallest value down to a power of ten and the largest up, and count the steps between them. The generator allows 1 to 6 cycles on each logarithmic axis.
How do I label a log axis?
Each decade is numbered 1 to 9, with some of the in-between numbers left out on short decades where they would touch; you supply the power of ten. Write the bottom value of each decade on its heavy line: 1, 10, 100, 1,000 for data that starts at 1, or 0.01, 0.1, 1, 10 for data that starts at 0.01. The 2 in the second decade then means 20, the 5 in the third means 500, and so on.
Where is zero on log paper?
There is none. A log axis gets closer to zero by a factor of ten with every decade but never reaches it, so zero and negative values can't be plotted. If your data includes zero, either use ordinary graph paper for that axis or plot the values that are greater than zero and note the rest.
Should I use semi-log or log-log paper?
Semi-log (one log axis) when one quantity changes by a constant factor for each equal step of the other, as in exponential growth and decay: those plot as straight lines. Log-log (both axes logarithmic) when y is proportional to a power of x, y = a·xb: those plot as straight lines with slope b. If you don't know which applies, plot a few points on each and see which comes out straight.
How do I find the slope on log-log paper?
From two points on the line, read their values and work out b = (log y₂ − log y₁) ÷ (log x₂ − log x₁). Measuring the line's rise and run with a ruler gives the same answer only when a decade is the same length on both axes. On the default 2 × 3-cycle Letter sheet they aren't (3.69 in across, 3.28 in up); choose 3 cycles across and 4 up and both decades are 2.46 in.