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MathidiaMathematics Academy
Ratio, variation and rates
Lesson 32 of 75
Easy practice

Linearising proportion

Objective. Transform proportional laws into straight-line relationships.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

Linearising proportion means transforming a nonlinear proportional law, such as y = kxⁿ, into a straight-line equation so the constants can be found from a graph. Taking logarithms of both sides gives log y = n log x + log k, which matches the straight-line form Y = mX + c with Y = log y, X = log x, gradient n and intercept log k. Plotting log y against log x therefore produces a straight line whose gradient reveals the power n.

Worked example

A relationship y = kxⁿ gives the linearised equation log y = n log x + log k. If a graph of log y against log x has gradient 3 and y-intercept 0.5, find n and k.

3 … 0.5
Coordinate grid with the line 3 … 0.5
  1. 1Compare log y = n log x + log k with the gradient-intercept form
  2. 2Gradient equals n, so n = 3
  3. 3Intercept equals log k, so log k = 0.5
  4. 4Solve for k: k = 10^0.5 ≈ 3.16

Answer: n = 3, k ≈ 3.16

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Common mistake. Students often plot the raw y against x expecting a straight line even when the relationship is a power law, missing that only the logged variables are linear. Always take logarithms of both variables first when testing for a power-law relationship.

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