LogPlot vs Linear Plot: When to Use Logarithmic ScalingUnderstanding how to display numerical data effectively is essential for clear communication in science, engineering, finance, and data journalism. Two of the most common ways to represent continuous numeric data on a graph are linear plots and logarithmic (log) plots. Choosing between them — or using both — depends on the data’s range, distribution, and the story you want the visualization to tell. This article explains the differences, when to use each type, how to interpret them, and practical tips to avoid common pitfalls.
What are linear and log plots?
A linear plot (linear scale) places equal spacing between equal absolute increments on an axis. For example, on a linear x-axis the distance between 0 and 10 is the same as between 90 and 100.
A logarithmic plot (log scale) spaces ticks according to the logarithm of the values. On a base-10 log axis, ticks might be at 1, 10, 100, 1000 — each tick is a constant multiplicative step. Log plots are useful when values span several orders of magnitude or when multiplicative relationships (e.g., exponential growth or power laws) are central.
Visual differences and interpretation
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Linear plot:
- Shows absolute differences directly: a change from 10 to 20 looks the same as 90 to 100.
- Straight lines indicate linear relationships (y = ax + b).
- Useful when additive differences are meaningful.
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Log plot:
- Compresses large values and expands small values so that multiplicative changes appear as additive distances.
- Exponential growth y = A e^{kt} becomes a straight line on a semilog plot (log y vs linear x).
- Power laws y = A x^{n} become straight lines on a log-log plot (log y vs log x).
- Zero and negative values cannot be plotted directly on a log scale.
Key fact: use a log axis when differences in orders of magnitude or multiplicative relationships matter.
When to use a linear plot
Use a linear scale when:
- Data values occupy a relatively narrow numeric range.
- Additive differences are important (e.g., temperature changes in degrees Celsius, absolute error).
- You need to preserve the perception of absolute differences for non-technical audiences.
- Data include zero or negative values that are meaningful and must be shown as-is.
Examples:
- Daily temperatures around a mean (e.g., 15–30°C).
- Heights of individuals.
- Measurement error analysis where absolute deviations matter.
When to use a log plot
Use a logarithmic scale when:
- Values span multiple orders of magnitude (e.g., 0.001 to 10,000).
- Data growth/decay is multiplicative or exponential (e.g., population growth, compound interest, radioactive decay).
- You want to detect power-law behavior or straight-line trends in log-log or semilog space.
- You need to visualize relative changes (percent changes) more naturally than absolute differences.
Examples:
- Earthquake magnitudes, frequency spectra, and acoustic intensity.
- Population growth over centuries.
- Financial returns when comparing multiplicative gains.
- Frequency distributions (histograms) where tail behavior spans orders of magnitude.
Types of log plots
- Semilog plot (log-linear): log scale on the y-axis, linear on the x-axis. Useful for exponential growth/decay: straight lines indicate y ∝ e^{kt}.
- Semilog-x: log scale on the x-axis, linear on the y-axis. Useful when input spans orders of magnitude but response changes additively.
- Log-log plot: log scale on both axes. Useful for detecting power laws: y ∝ x^{n} appears as a straight line whose slope equals n in log space.
Practical considerations and preprocessing
- Handling zeros and negatives:
- Logarithms are undefined for zero or negative values. Options: offset data by adding a constant (careful — this changes interpretation), remove zeros, plot absolute values with a sign indicator, or use a symmetric log transform (symlog) that behaves linearly near zero and logarithmically away from it.
- Choice of base:
- Base 10 is common for orders of magnitude; base e (natural log) or base 2 have their own interpretive advantages. The visual spacing is the same regardless of base; only tick labels differ.
- Tick labeling:
- Use readable ticks: show powers of ten on base-10 logs, and avoid too many intermediate ticks. Clearly label axis scale (e.g., “log10(y)” or indicate units).
- Perception:
- Nontechnical audiences may misinterpret compressed large values on log scales. Add annotations or show both linear and log versions when necessary.
- Statistical summaries:
- Means, variances, and regression interpretations differ in log space (e.g., arithmetic mean on log-transformed data corresponds to geometric mean on original scale). Be explicit about the scale used for analyses.
Examples and diagnostics
- Exponential growth: plotting population vs time on semilog-y. If points line up, growth rate is roughly constant; slope gives k in y = A e^{kt}.
- Power-law behavior: plotting frequency vs magnitude on log-log. A straight line indicates power-law; slope gives exponent.
- Mixture data: if small values are important (e.g., noise floor) but you also need to show large values, consider symlog or plotting two panels (linear and log) for clarity.
Common pitfalls
- Hiding important structure: log plots can hide additive variation at high values.
- Misleading viewers: failing to indicate the axis is logarithmic can cause misinterpretation.
- Incorrect transformations: applying log to data with zeros or negative values without clear justification distorts results.
- Over-reliance: Not every skewed dataset needs a log transform; consider alternatives (e.g., square-root transform, quantile plots, or nonparametric methods).
Quick decision checklist
- Do values span several orders of magnitude? — Yes → consider log.
- Is the relationship multiplicative/exponential or a power law? — Yes → log (semilog or log-log).
- Are zero/negative values essential? — Yes → avoid pure log or use symlog/other transforms.
- Is the audience likely unfamiliar with logs? — Consider showing both scales or adding explanations.
Short examples (conceptual)
- Semilog-y: Plotting COVID-19 cases vs. days often used early in outbreaks; exponential growth appears as a straight line.
- Log-log: Plotting city population rank vs. population size to check Zipf’s law; straight line suggests power-law scaling.
- Linear: Plotting daily temperature over a month when values vary within tens of degrees.
Conclusion
Logarithmic and linear plots are complementary tools. Use linear scales when absolute differences matter and ranges are modest. Use log scales when multiplicative changes, percentages, or wide dynamic ranges are central. Always label axes, explain transformations, and choose the scale that most faithfully communicates the phenomenon you’re analyzing.
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