In the foregoing problem, we saw how the use of logarithmic scales made it possible to display information over an extremely large range of values. The next two problems show another use for logarithmic scales, that of fitting a mathematical function to experimental data.
PROBLEM 4. Very high energy particles (electrons and protons) are found in the radiation belts of some planets (e.g., Earth, Jupiter, Saturn), and a plot of the number of particles found at different energies is called a "spectrum." Often the spectrum has a shape that can be represented by an equation of the form N = KE^m where N is the number of particles at a certain energy, E; K is a proportionality factor; and m is called the spectral index.
When the spectrum has such a shape, we call it a power-law spectrum, and the experimenter studying such a spectrum wants to know the values of m and K. Table 6.1 shows values of N measured at several Es during the flight of Pioneer 10 past Jupiter. For these data, find the best value of m and of K. (N is really the number of particles hitting a detector per unit time, or the counting rate, which is why the number can be a fraction.)
Solution: Using logarithms on the expression N = KE^m results in log N = log K + m log E, or, to obtain the form of a linear equation y = mx + b, log N = m log E + log K.
We can find logarithms for the values of E and N in the table (or we can use log-log graph paper and circumvent this step), plot the points, and draw the best straight line through this set of points. Then m will be the slope of the line, and K will be the value of N for which log E = 0. (Note that this is the value that lies on the best straight line, and not necessarily any value in the data set.)
We observe in Fig. 6.2 that the intercept on the log N scale is 3.5. Since log N = 3.5 when log E = 0, we have log K = 3.5, so K = 10^3.5 = 3200. The points (0, 3.5) and (1.0, 0) are on the best fit line, so m = (3.5 - 0)/(0 - 1.0) = - 3.5. So N = (3200)E^-3.5 .