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14.8PureStretch

Logarithms and Non-Linear Data

Taking logs turns curved relationships into straight lines. Plotting log values lets you fit power and exponential models to real data and read off their constants from the gradient and intercept.

30 min Exponentials and Logarithms
Edexcel AS Level Maths: 14.8 Logarithms and Non-Linear DataCurated by Kepler, not produced by us. Watch it before the notes below.
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What you'll be able to do

  • Linearise y = axⁿ by taking logs
  • Linearise y = abˣ by taking logs
  • Read constants from the gradient and intercept
  • Choose the right log-plot for a model
1

Power law: y = axⁿ

Taking logs of y=axny = ax^n gives a straight line when you plot log⁡y\log y against log⁡x\log x. The gradient is the power nn and the intercept is log⁡a\log a.

log⁡y=nlog⁡x+log⁡a\log y = n\log x + \log a
Plot log⁡y\log y vs log⁡x\log x: gradient nn, intercept log⁡a\log a.
2

Exponential law: y = abˣ

For y=abxy = ab^x, take logs to get a line when plotting log⁡y\log y against x\textbf{x} (not log⁡x\log x). The gradient is log⁡b\log b and the intercept is log⁡a\log a.

log⁡y=(log⁡b)x+log⁡a\log y = (\log b)x + \log a
Plot log⁡y\log y vs xx: gradient log⁡b\log b, intercept log⁡a\log a.

Tip — Power law → log y vs log x. Exponential law → log y vs x. The choice of axes is the key decision.

3

Reading the constants

Once you have the straight-line gradient and intercept, reverse the logs to recover the original constants (aa, nn or bb).

Example 1A log y vs x plot has gradient log⁡b\log b. If the gradient is 0.30100.3010, find bb.
1log⁡b=0.3010\log b = 0.3010, so b=100.3010b = 10^{0.3010}.
Answerb=2b = 2

Formula recap

log⁡y=nlog⁡x+log⁡a\log y = n\log x + \log a
Power law (log y vs log x).
log⁡y=(log⁡b)x+log⁡a\log y = (\log b)x + \log a
Exponential law (log y vs x).
intercept=log⁡a, gradient=n or log⁡b\text{intercept} = \log a, \ \text{gradient} = n \text{ or } \log b
Reading constants.

Common mistakes to avoid

Plotting log y vs log x for an exponential model.
Exponential y = abˣ ⟶ plot log y vs x (only x is logged for power laws).
Forgetting to undo the log to find a or b.
The intercept is log a, so a = 10^(intercept).

Key takeaways

  • Power law y = axⁿ: log y vs log x is linear (gradient n, intercept log a).
  • Exponential y = abˣ: log y vs x is linear (gradient log b, intercept log a).
  • Undo the logs to recover the original constants.

Test yourself

Answer a few questions, then review anything you missed.

Next step

Check your understanding

Work through questions on Logarithms and Non-Linear Data. Use the feedback to decide what needs another look.

8 past-paper questions · 25 marks

Practise this topic

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