Logarithmic × logarithmic
Log-log
graph paper
Both axes logarithmic. Built for power-law relationships, frequency response analysis, and any data spanning multiple orders of magnitude on both variables.
About log-log paper
Both axes are logarithmic
Log-log graph paper (also called logarithmic paper or full-log paper) has both the horizontal and vertical axes on logarithmic scales. On each axis, equal distances represent equal ratios — the space from 1 to 10 equals the space from 10 to 100 equals the space from 100 to 1,000.
The key property: any power-law relationship of the form y = axⁿ becomes a perfectly straight line on log-log paper. The slope of that line equals the exponent n, which is a fundamental physical or statistical constant in many disciplines.
Log-log paper is the standard tool for identifying and quantifying power laws in data — a task that is otherwise very difficult to do by visual inspection on ordinary graph paper.
The log-log test: If data forms a straight line on log-log paper, the relationship is a power law (y = axⁿ). The slope of the line equals the exponent n.
Log-log paper cannot display zero or negative values — the logarithm of zero is undefined. Both variables must be strictly positive.
| vs semi-log | Log-log | Semi-log |
|---|---|---|
| Axes | Both logarithmic | One linear, one log |
| Straight line means | Power law y = axⁿ | Exponential y = a·bˣ |
| Slope equals | Exponent n | Log of growth rate |
| Example | Kepler's law | Bacterial growth |
Applications
What is log-log paper used for?
Physics — scaling laws
Physical scaling laws (how heat transfer, drag force, or structural strength scale with size) are power laws. Kepler's third law (T² ∝ r³) and Stefan-Boltzmann law both appear as straight lines on log-log paper.
Engineering — impedance plots
Impedance vs. frequency plots where both gain and frequency span many decades use log-log axes. True impedance magnitude plots require log-log scales where both axes cover multiple orders of magnitude.
Ecology — species-area law
The species-area relationship (S = cA𝑧) is a power law. Ecologists use log-log paper to determine the exponent z — a fundamental constant in island biogeography, typically between 0.2 and 0.35.
Seismology — Gutenberg-Richter
The frequency of earthquakes of different magnitudes follows the Gutenberg-Richter power law. Seismologists plot earthquake frequency vs. magnitude on log-log scales to determine the b-value of a region.
Materials — fatigue and fracture
Paris's law for fatigue crack growth (da/dN = C·ΔK𝒟) is a power law. Log-log paper is used to plot crack growth rate vs. stress intensity factor range to determine the Paris exponent m.
Astronomy — luminosity relations
The mass-luminosity relationship in stellar astrophysics (L ∝ M𝐺) and the period-luminosity relation in Cepheid variables are power laws naturally represented on log-log axes.
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FAQ
Frequently asked questions
What is log-log graph paper?+
What is the difference between log-log and semi-log paper?+
Can log-log paper show negative numbers or zero?+
How do I read the slope of a line on log-log paper?+
Is logarithmic graph paper the same as log-log paper?+
Related types
Other graph paper formats
Semi-log paper
One logarithmic axis — for exponential growth data where only one variable scales non-linearly.
Cartesian paper
Linear-linear grid — use alongside log-log paper when comparing scaled and raw data.
Engineering paper
Dual-grid for precision field data collection at major and sub-increment scales simultaneously.
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