When we take measurements of the same general type, a power law of the form
\(y=\alpha x^{\beta}\) 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 \(x>0\) and all \(y>0\).
Suppose we have data pairs \((x, y)\) and we want to find constants \(\alpha\) and
\(\beta\) such that \(y=\alpha x^{\beta}\) is a good fit to the data. First, make
the logarithmic transformations \(x^{\prime}=\log x\) and \(y^{\prime}=\log y .\)
Next, use the \(\left(x^{\prime}, y^{\prime}\right)\) data pairs and a
calculator with linear regression keys to obtain the least-squares equation
\(y^{\prime}=a+b x^{\prime} .\) Note that the equation \(y^{\prime}=a+b
x^{\prime}\) is the same as \(\log y=a+b(\log x)\). If we raise both sides of
this equation to the power 10 and use some college algebra, we get
\(y=10^{a}(x)^{b} .\) In other words, for the power law model \(y=\alpha
x^{\beta}\), we have \(\alpha \approx 10^{a}\) and \(\beta \approx b\).
In the electronic design of a cell phone circuit, the buildup of electric
current (Amps) is an important function of time (microseconds). Let \(x=\) time
in microseconds and let \(y=\) Amps built up in the circuit at time \(x\).
$$
\begin{array}{c|ccccc}
\hline x & 2 & 4 & 6 & 8 & 10 \\
\hline y & 1.81 & 2.90 & 3.20 & 3.68 & 4.11 \\
\hline
\end{array}
$$
(a) Make the logarithmic transformations \(x^{\prime}=\log x\) and
\(y^{\prime}=\log y .\) Then make a scatter plot of the \(\left(x^{\prime},
y^{\prime}\right)\) values. Does a linear equation seem to be a good fit to
this plot?
(b) Use the \(\left(x^{\prime}, y^{\prime}\right)\) data points and a calculator
with regression keys to find the least-squares equation \(y^{\prime}=a+b
x^{\prime} .\) What is the correlation coefficient?
(c) Use the results of part (b) to find estimates for \(\alpha\) and \(\beta\) in
the power law \(y=\) \(\alpha x^{\beta .}\) Write the power law giving the
relationship between time and Amp buildup.
Note: The TI-84Plus calculator fully supports the power law model. Place the
original \(x\) data in list \(\mathrm{L} 1\) and the corresponding \(y\) data in
list \(\mathrm{L} 2 .\) Then press STAT, followed by CALC, and scroll down to
option A: PwrReg. The output gives values for \(\alpha, \beta\), and the
correlation coefficient \(r\).
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