Logarithmic paper is the tool that turns curves into straight lines. When a process multiplies — bacteria doubling, interest compounding, sound decaying — a linear plot shows an ever-steepening curve that hides the rate. On log paper the same data draws a straight line whose slope is the rate. This guide covers reading and plotting on the semi-log and log-log sheets the generator prints.

Anatomy of a log axis

A log axis is built from repeated blocks called decades. Within each decade, nine lines mark log₁₀(1) through log₁₀(9); the heavy line at the bottom of each decade is the power of ten itself. Visually:

• The 1 line (heavy) sits at the decade’s base — that’s 1, 10, 100, 1000… depending on which decade.
• Lines 2–9 pack progressively tighter toward the top of the decade.
• The next heavy line starts the next decade at 10× the previous one.

A 3-decade axis spans a thousand-fold range. The bunching isn’t a defect — it’s the true shape of the logarithm: the gap between 1 and 2 (a doubling) is the biggest gap in the decade, and 8-to-9 (a 12.5% step) is the smallest.

Semi-log vs log-log

Semi-log (one log axis, one linear) is for data where one variable grows multiplicatively. Classic uses:

• Population or bacterial growth vs time
• Compound interest vs years
• Radioactive decay / capacitor discharge vs time
• pH, decibels, Richter — already-logarithmic quantities vs linear ones
• Anything where you care about percentage change rather than absolute change

Convention puts the log on the Y axis (the measured quantity) and linear time on X — that’s the default semi-log mode. Flip to log-on-X when the independent variable is the one spanning decades (frequency response plots, for instance — audio filters are drawn log-frequency × linear-dB).

Log-log makes both axes decades. It’s for power laws: data of the form y = a·xᵏ. On log-log paper, y = a·xᵏ becomes a straight line with slope k — which is why every physics paper on scaling uses it. Examples that famously straighten on log-log: metabolic rate vs body mass (k ≈ 0.75), Zipf’s law for word frequencies, earthquake magnitude vs frequency, and any allometric scaling.

Choosing the decade count

Set decades to the number of powers of ten your data needs on the log axis:

• Data from 5 to 800 → needs to cover 10⁰–10³ → 3 decades
• Data from 0.1 to 50 → 10⁻¹ to 10² → 3 decades (the paper labels 1–1000 but the decades are position-identical — read them as your own powers)
• Data from 20 to 200 → fits inside one decade, but a single decade wastes resolution → still pick 1 decade? No — pick the count that spans your range: 20–200 crosses 10¹–10³ → 2 decades minimum, and the sheet will place your data in the upper bands

The rule: decades = ⌈log₁₀(max)⌉ − ⌊log₁₀(min)⌋, rounded up. If your data doesn’t start near a power of ten, it just occupies the upper portion of the first decade — that’s normal.

Reading and plotting, worked example

Plot bacterial colony counts — 90, 340, 1200, 4600 cells at hours 0, 1, 2, 3 — on semi-log-Y paper with 4 decades (the generator default labels decades 10⁰–10⁴, i.e. 1–10000):

1. Locate each count’s decade: 90 is in decade 2 (10–100); 340 and 1200 in decade 3; 4600 in decade 4.
2. Position within the decade: 90 → log₁₀(90) ≈ 1.95, so 95% up decade 2 — just under the heavy 100 line. 340 → log₁₀(340) ≈ 2.53, so 53% up decade 3 — between the 3-line and 4-line, nearer 3. 1200 → log₁₀(1200) ≈ 3.08 — just above the heavy line of decade 4. 4600 → log₁₀(4600) ≈ 3.66 — between the 4 and 5 lines of decade 4.
3. Plot against linear X (hours, evenly spaced).
4. Connect the dots: if the culture grows exponentially the four points make a straight line — and its slope (rise in decades per hour) is the growth rate: roughly 0.57 decades/hour here, i.e. doubling every ~0.53 h.

That slope is the prize. On linear paper you’d have a curve shooting upward and no direct rate; on log paper the rate is the line’s angle.

Reading slopes as real quantities

On semi-log, slope = decades per x-unit. Multiply by ln(10) ≈ 2.303 to get the continuous rate constant r in y = y₀·e^(rt). For doubling time: t₂ = log₁₀(2)/slope ≈ 0.301/slope.

On log-log, slope = the exponent. Measure it by counting decades: pick two points where the line crosses decade-grid intersections, slope = Δlogy/Δlogx = (decades risen)/(decades run). A slope of 2 means quadratic scaling; −1 means inverse proportion; 0.75 means biology’s famous Kleiber’s law.

Gotchas that bite first-timers

• Zero and negatives can’t go on a log axis — offset the data or keep a linear axis.
• Don’t interpolate linearly inside a decade — halfway between the 2-line and 3-line is not 2.5; it’s about 2.45 (log₁₀ interpolation). Read against the printed 1–9 lines.
• Averaging log-axis values is not arithmetic — the visual midpoint between two values is their geometric mean (√(a·b)), not their average. That’s usually what you want on multiplicative data anyway.
• Decade labels on printed paper are relative — if your data is 0.001–0.5, the sheet’s “1” line can stand for 0.001; what matters is each decade = ×10.

Print a semi-log sheet with 3 decades and re-plot a dataset you know is exponential. The moment the curve snaps straight is the moment log paper makes permanent sense.

Frequently asked questions

What does a 'decade' mean on log paper?

One power of ten — the distance from 1 to 10, or 10 to 100, or 100 to 1000. A '3-decade' log axis spans three such blocks (e.g. 1–1000). Each decade occupies the same physical height on the paper, which is what makes multiplicative growth plot as a straight line.

Why do the lines bunch up inside each decade?

Because log spacing maps position to log₁₀(value), and log₁₀ is not uniform: log₁₀(2) ≈ 0.30, log₁₀(5) ≈ 0.70, log₁₀(9) ≈ 0.95. So the 2 sits 30% up the decade and the 9 crowds the top. The bunching is the honest shape of a logarithm — you read intermediate values by interpolating between the 1–9 lines, never by assuming equal spacing.

How do I tell if my data needs log paper?

Two quick tests. First: does your data span more than one order of magnitude (smallest to largest value ×10 or more)? Second: is the process multiplicative — percentages, doubling times, growth rates? If either is yes, plot on semi-log and watch the curve straighten.

What does a straight line mean on each kind of paper?

On linear paper: constant addition (linear growth). On semi-log: constant multiplication — exponential growth or decay, and the slope gives the rate directly. On log-log: a power law y = a·xᵏ, where the slope is the exponent k. Straight lines on log axes are how you spot the underlying law.

How do I plot a value like 47 on log paper?

Find the decade containing it (10–100), then its position inside: log₁₀(47) ≈ 1.67, so 67% of the way up the second decade — just above the 4-line and below the 5-line, closer to 5. You never measure '47 mm from the axis'; you always locate by decade-then-interpolation.

Can negative or zero values go on a log axis?

No — log₁₀(0) is −∞ and log of negatives is undefined. If your data includes zero or negatives you need a different treatment: offset the data (add a constant), use a linear axis, or use a signed log variant like symlog (not what this paper draws). The printed axis always starts at 10⁰ = 1.