Log, semi-log and log-log paper
Guide · Updated 2026-08-04 · 3 min read
Logarithmic graph paper looks intimidating — the lines bunch up and repeat in a strange rhythm — but it exists to do one very useful thing: turn certain curves into straight lines you can actually read. This guide explains logarithmic scales and decades, then shows when to reach for semi-log paper versus log-log paper.
Linear versus logarithmic scales
On an ordinary linear axis, equal steps cover equal amounts: 1, 2, 3, 4. On a logarithmic axis, equal steps cover equal multiples: 1, 10, 100, 1000. Each of those tenfold jumps is called a decade, and within a decade the gridlines for 2, 3, 4… up to 9 crowd closer and closer together before the next power of ten. That uneven spacing is not a printing quirk — it is the whole idea. A log scale lets one sheet hold values that span several orders of magnitude, from 1 to 10,000, without the small numbers vanishing into the corner.
Semi-log paper: one log axis
Semi-log paper is logarithmic on one axis (usually the vertical) and linear on the other. Its speciality is exponential data — anything that grows or decays by a constant percentage per step. Bacterial growth, radioactive decay, compound interest and the early spread of an epidemic all follow this pattern, and on ordinary paper they draw as steep curves that are hard to compare. On semi-log paper the same data plots as a straight line, and the slope of that line tells you the rate directly. Straight means “steady exponential”; a bend means the rate is changing.
To set it up, count how many powers of ten your data covers and choose that many decades: values from 10 to 10,000 span three decades. Put time or the evenly spaced variable on the linear axis. Reach for a semi-log sheet when exactly one of your quantities changes multiplicatively.
Log-log paper: both axes log
Log-log paper is logarithmic on both axes. It is built for power laws — relationships of the form y = a·xⁿ, which appear all over science and engineering, from area-and-volume scaling to frequency response and many empirical calibration curves. A power law is a curve on linear paper but a straight line on log-log paper, and here the slope of the line equals the exponent n. Reading an exponent off a graph by measuring a slope is a genuinely handy trick.
Choose log-log when both variables range over wide scales, or when you suspect a power-law relationship and want to test it. Choose semi-log when only one variable explodes exponentially. A quick way to remember it: semi-log for growth and decay over time, log-log for scaling laws. Generate one on the log-log page.
Reading values between the lines
The commonest mistake on log paper is interpolating as if the gridlines were evenly spaced. They are not — within a decade the gap between 1 and 2 is far wider than the gap between 8 and 9. Read each value against the printed line nearest to it, and remember that halfway up a decade is not the value 5; because the scale is logarithmic, the midpoint of a 1-to-10 decade sits at about 3.16 (the square root of ten). Keeping that in mind stops a lot of plotting errors.
A related cousin: probability paper
Logarithmic paper is one member of a family of “straightening” grids that reshape an axis so a particular pattern becomes a line. Another is probability paper, whose vertical axis is stretched so that normally distributed data plots straight — a quick pencil-and-paper check for a bell curve, with the mean read at the 50% line. If log paper makes sense to you, probability paper will feel familiar.
In short
- Data grows or shrinks by a fixed percentage? Semi-log — the line’s slope is the rate.
- Two quantities related by a power? Log-log — the slope is the exponent.
- Checking for a normal distribution? Probability paper.
Set the number of decades to cover your range, plot as usual, and let the straight line do the interpreting. When you print, use the true-size settings so the decades stay proportioned correctly.