Simulic
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Log scales and linearisation – semi-log and log–log graphs

MathematicsFunctions & GraphsAges 16–17

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Plot real data on ordinary axes, semi-log axes (log y against x) and log–log axes (log y against log x): y = a·bˣ becomes a straight line on semi-log axes and y = a·xⁿ on log–log axes. Read the gradient and intercept to find a, b and n, drag your own line and check the residuals. The Log scales screen compares earthquake magnitude, sound level in decibels and pH, where equal steps mean equal ratios.

Lesson: Logarithms: log scales, linearising y = a·bˣ and y = a·xⁿ, the Richter, decibel and pH scales

What it shows

Taking logarithms turns two common curves into straight lines. If y = a·bˣ, then log y = log a + x·log b, so log y against x is a line with gradient log b and intercept log a. If y = a·xⁿ, then log y = log a + n·log x, so log y against log x is a line with gradient n. The data sets cover radioactive decay, bacterial growth, earthquake frequency, planetary orbits, a pendulum and animal metabolism. Logs are base 10 and the line is fitted to the logged values. Decay, bacteria and pendulum data are simulated with measurement noise.

How to use

Choose a Data set, then press y – x, log y – x or log y – log x and watch whether the points straighten. Read the gradient triangle and the results panel for a, b or n. Press Drag and move the two orange circles to place your own line, and tick Residuals to check for a pattern. On Log scales, drag A and B along the scale.

Parameters you can change

  • Screen Linearise, Log scales
  • Data set Radioactive decay of iodine-131, Bacterial growth, Earthquakes per year by magnitude, Orbital periods of the planets, Period of a simple pendulum, Metabolic rate of mammals
  • Axes y against x (ordinary axes), log y against x (semi-log), log y against log x (log–log)
  • Line Least squares, Drag your own line
  • Show residual plot
  • Log scale (Log scales screen) Earthquake magnitude (Richter), Sound intensity level (dB), pH

Questions to explore

  1. On which axes do the iodine-131 data lie on a straight line, and what half-life does the gradient give?
  2. What does the gradient of the log–log graph for the planets tell you about Kepler's third law?
  3. How many times more intense is a 90 dB sound than a 60 dB sound, and why is it not 1.5 times?

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