Question C.1.1: Find a functional description of the following data:...

These data do not lie close to a straight line when plotted on linear or semilog axes. However, they do when plotted on log-log axes. Thus a power function y = bx^m can describe the data. Using the transformations X = log x and Y = log y, we obtain the new data table:

From this table we obtain

\sum\limits_{i=1}^4 X_i=1.3803 \quad \sum\limits_{i=1}^4 Y_i=5.5525

\sum\limits_{i=1}^4 X_i Y_i=2.3208 \quad \sum\limits_{i=1}^4 X_i^2=0.6807

Using X, Y , and B = log b instead of x, y, and b in (C.1.1) and (C.1.2) we obtain

m \sum\limits_{i=1}^n x_i^2+b \sum\limits_{i=1}^n x_i=\sum\limits_{i=1}^n y_i x_i                  (C.1.1)

m \sum\limits_{i=1}^n x_i+b n=\sum\limits_{i=1}^n y_i            (C.1.2)

0.6807m + 1.3803B = 2.3208

1.3803m + 4B = 5.5525

The solution is m = 1.9802 and B = 0.7048. This gives b = 10^B = 5.068. Thus, the desired function is y = 5.068x^{1.9802}.