Activities

Logarithms - modelling with the power function y = Axn and log-log graphs

For some real life data, a power function y = Axn may be an appropriate model.

This is confirmed if the graph of log y against log x is close to a straight line (log-log graph).

Comparing the equation with the gradient-intercept form (y = mx + c):

  • n is the gradient of the straight line
  • log A is the intercept on the vertical axis.

    • The time that each planet takes to orbit the sun increases the further the planet is from the sun.

    (The orbit is an ellipse. The sun is at a focus and not at the centre.)

    The relationship is certainly not linear as shown by the data and graph.
     

    The log-log graph does show a strong linear relationship. Fitting a straight line gives:

    x (million miles) is the greatest distance of the planet from the centre of its orbit.

    y (years) is the time that the planet takes to orbit the sun.

    Compare the times given by the model with the actual data. The fit is excellent.
     

    x
    y
    value using model
    36
    0.241
    0.241
    67
    0.615
    0.612
    93
    1
    1.00
    142
    1.88
    1.89
    483
    11.9
    11.8
    886
    29.5
    29.4
    1782
    84
    84.0
    2793
    165
    165
    3670
    248
    248

    The regression features of a graphic calculator or software could also be used to determine the equation and the suitability of the model.