2.6.3 Power Functions and Log-Log Plotting

We turn now to data of another sort, namely, (simulated) data giving the position of a falling body at half-second time intervals. Table 3 contains a refinement of the falling body data in Table 1 of Section 2.1, where we explored a theoretical model for these data from elementary physics: $s=c{t}^2,$ where \(s\) is distance fallen at time \(t\) and \(c\) is a constant. We call such a function - a constant multiple of a constant power of the independent variable - a power function. This is to distinguish such functions from exponential functions, which have constant base and variable exponent. We can now address the question of whether the data actually fit a power function model.

Activity 7

1. Take logs of both sides of the proposed functional relationship $s=c{t}^2$, and simplify as much as possible. (You decide what base to use for the logarithms.)

2. The resulting equation says something is a linear function of something else. What is a linear function of what? What is the slope of that linear function?

3. Propose a graphical test for deciding whether the data in Table 3 fit the theoretical model.

4. Use the Plot Data tool to carry out a test of whether these data fit the model. The data from Table 3 are already entered in the plotting tool, and you can choose whether to plot the points on an "ordinary" (Cartesian) scale, a semilog scale, or a "log-log" scale, that is, with logarithmic scales in both directions.

5. What do you conclude about whether the data fit the model?

Comment on Activity 7

A plot in which both horizontal and vertical scales are treated logarithmically, such as the one created in Activity 7, is called a log-log plot or just a log plot.