Log-Log Graph Paper (Printable PDF)

About this tool

Log-log graph paper has a logarithmic scale on both the horizontal and vertical axes, instead of the evenly spaced (linear) scale of ordinary graph paper. On a log scale the distance from 1 to 10 is the same as from 10 to 100 and from 100 to 1000 — each of those spans is one "decade." Inside a decade the minor lines (2, 3, 4 … 9) are spaced logarithmically, crowding closer together as the numbers grow. This lets you plot data that ranges across many orders of magnitude on a single sheet.

Worked example

Log-log paper turns power-law relationships into straight lines. If two quantities follow y = a·xⁿ — say, how the area of a square grows with its side length — plotting them on log-log axes gives a straight line whose slope equals the exponent n. Instead of a curve that is hard to read, you get a line you can measure: pick two points on it and the slope tells you the power. That is the classic reason scientists and engineers reach for log-log paper.

Key / conventions

Each axis is divided into decades — factor-of-ten spans (1–10, 10–100, and so on). The heavy lines mark decade boundaries; the lighter lines inside each decade mark 2 through 9, placed by their logarithm, so they bunch toward the top of the decade. To read a position, find which decade you are in from the heavy lines, then locate the minor line. Choose the number of decades to match the range your data covers on each axis.

FAQ

When should I use log-log paper instead of regular graph paper?

Use it when your data spans several orders of magnitude — where a linear grid would bunch the small values into a corner — or when you expect a power-law relationship and want to check it. If both variables stay within a single order of magnitude, ordinary graph paper is usually clearer.

What's the difference between log-log and semi-log paper?

Log-log paper is logarithmic on both axes. Semi-log paper is logarithmic on only one axis and linear on the other. Semi-log is used for exponential growth or decay, which becomes a straight line there; log-log is used for power laws.

How do I plot a power-law relationship?

Plot your (x, y) pairs directly on the log-log grid. If the relationship is a power law, y = a·xⁿ, the points fall along a straight line. The line’s slope is the exponent n, and you can read the constant a and the exponent n straight off the graph — the main advantage over a linear plot.

How many decades do I need?

Count the orders of magnitude your data covers on each axis. If x runs from about 1 to 1000, that is three decades; if y runs from 0.1 to 100, that is also three. Pick a sheet with at least that many decades per axis so all your points fit.