Family 09 · log / semi-log
Logarithmic graph paper with true subdivisions
Logarithmic graph paper spaces its lines by ratio, not by distance — each decade is one equal step, and the 2, 3, 4…9 subdivisions inside it sit at their exact log(m) positions. Choose semi-log or log-log, base 10 or base 2, and the number of decade cycles, then download a vector PDF whose subdivisions are mathematically generated, not evenly faked.
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Log / semi-log — ready to print
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The subdivisions are the whole point
On a logarithmic axis the position of the line for value m within a decade is proportional to log(m) / log(base). For base 10 that puts the “2” line at 30.1% of the way across the decade, the “3” at 47.7%, the “9” at 95.4% — bunched tight against the next decade boundary. That crowding is not a defect; it is the definition of the scale, and it is exactly what lets an exponential plot as a straight line.
This is where every cheap generator fails: they space 1 through 9 evenly, producing a linear grid that merely looks busy. Plotted on it, your exponential stays a curve and your power law bends. Here every subdivision is computed from the true logarithm, the decade boundaries carry a heavier weight and optional labels, and the geometry contract on the methodology page documents the exact formula and version.
Which log configuration should I choose?
| Configuration | Best for |
|---|---|
| Semi-log X · base 10 · 3 cycles | One logarithmic axis (10³ range) against a linear axis — the standard for exponential decay, frequency response and concentration curves. |
| Semi-log X · base 10 · 4 cycles | Four decades (10⁴ range) for wider data spans — electronics Bode plots, earthquake magnitudes, broad dilution series. |
| Log-log · base 10 · 3 cycles | Both axes logarithmic — power laws, allometric scaling and fractal relationships plot as straight lines. |
| Semi-log X · base 2 · 5 cycles | Binary decades for computer-science data — algorithm complexity, memory sizes and anything that doubles. |
“Cycles” is the number of decades: 3 cycles = a 10³ (thousand-fold) range on a base-10 axis.
What is logarithmic paper used for?
Anywhere data spans orders of magnitude or follows an exponential or power law. Electrical engineers plot frequency response (Bode plots) on semi-log paper; physicists chart radioactive decay and capacitor discharge; biologists graph bacterial growth and dose-response curves; seismologists work in logarithmic magnitudes. On log-log paper, power-law relationships — metabolic scaling, city-size distributions, fractal measurements — collapse to straight lines whose slope is the exponent.
For data that fits a modest range, ordinary graph paper is clearer; for direction-based data see polar paper, and for technical sketches with a major/minor grid, engineering paper.
How do I print log paper at true scale?
Log paper is the least forgiving of mis-scaling, because a uniform shrink changes the apparent position of every subdivision and silently corrupts the values you read off. Print at 100% scale — never “Fit to page” — and verify a full decade: the distance from the 1 line to the 10 line is one cycle width, and ten of them must match the printed page width the generator reports.
Never distort a mathematical scale to “fit”
Rescaling a log axis does not just shrink it — it changes the ratios the scale encodes. Always print at 100% and, if your printer drifts, apply a uniform correction rather than letting the dialog rescale non-uniformly.
Frequently asked questions
What is logarithmic graph paper?
Logarithmic graph paper spaces its lines by logarithm rather than by equal distance, so each fixed step represents multiplication by a constant (the base) instead of addition. On base-10 paper a line is placed at 1, 2, 3 … 9, 10, 20, 30 … — the gap from 1 to 2 equals the gap from 10 to 20, because both are a ratio of 2. This compresses data that spans many orders of magnitude onto one sheet and turns exponential curves into straight lines you can read and extrapolate.
Why are the subdivisions unevenly spaced?
Because that is what logarithmic means. Within each decade the 2, 3, 4 … 9 lines crowd closer together as they approach the next decade — the position of the line for value m is proportional to log(m). A generator that spaces 1–9 evenly is not log paper at all; it is a linear grid mislabelled, and it will plot your exponential as a curve when it should be a line. This generator computes every subdivision at its true log(m)/log(base) position, which is the single feature cheap generators get wrong.
Semi-log or log-log?
Semi-log makes one axis logarithmic and leaves the other linear: use it when one quantity is exponential (decay, growth, frequency response) so it plots as a straight line against the linear variable. Log-log makes both axes logarithmic: use it for power-law relationships (y = a·xⁿ), which then also plot as straight lines whose slope is the exponent n. Choose the number of cycles to cover the range of your data — three cycles is 10³, a thousand-fold span.