Expand Your Knowledge: Logarithmic Transformations, Power Law Model When we take measurements of the same general type, a power law of the form y = axb often gives an excellent fit to the data. A lot of research has been conducted as to why power laws work so well in business, economics, biology, ecology, medicine, engineering, social science, and so on. Let us just say that if you do not have a good straight-line fit to data pairs (x, y), and the scatter plot does not rise dramatically (as in exponential growth), then a power law is often a good choice. College algebra can be used to show that power law models become linear when we apply logarithmic transformations to both variables. To see how this is done, please read on. Note: For power law models, we assume all x > 0and all y > 0.Suppose we have data pairs (x, y) and we want to find constants a and b suchthat y = axbis a good fit to the data. First, make the logarithmic transformations x’ = log x and y’ = log y. Next, use the (x’, y’) data pairs and a calculator with linear regression keys to obtain the least-squares equation y’ = a + bx’.
x | 2 | 4 | 6 | 8 | 10 |
y | 1.81 | 2.9 | 3.2 | 3.68 | 4.11 |
(a) Make the logarithmic transformations x’ = log x and y’ = log y. Thenmake a scatter plot of the values. Does a linear equation seem to be a good fit to this plot?
(b) Use the (x’, y’) data points and a calculator with regression keys to find the least-squares equation y’ = a + bx’. What is the sample correlation coefficient?
(c) Use the results of part (b) to find estimates for a and b in the power law y = axb. Write the power law giving the relationship between time and Amp buildup.
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