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, 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 $x .$ (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 leastsquares equation $y^{\prime}=a+b x^{\prime} .$ What is the sample 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/TI-83Plus/TI-nspire calculators fully support the power law model. Place the original $x$ data in list $L 1$ and the corresponding $y$ data in list $L 2 .$ Then press STAT, followed by CALC, and scroll down to option $\mathbf{A}:$ Pwr Reg. The output gives values for $\alpha, \boldsymbol{\beta},$ and the sample correlation coefficient $r$.
