GRAPHING ON A COMMON LOG SCALE

The graph of the log of p as a function of the log of q has utility.  If the plot of log p vs. log q is a straight line, then p(q) is a simple power function.  A simple power function is a monomial or an expression with only one exponent.  It will be in the form of p = kq where k is the proportionality constant derived from the graph’s intercept and n is the power or log derived from the graph’s slope.  If the plot of log p vs. log q is not a straight line, then p(q) is not a simple power function.  Many relationships between two physical variables are not linear; however some are simple power functions.  

An example of a simple power function relationship is the fluid flow through a tube.  Fluid flow can be expressed by f = (h/l) d, where f is the flow rate, h is the height difference between the ends of the tube, l is the length of the tube, and d is the diameter of the tube.  If we plotted f vs. d on a standard plot, we would see a steeply rising curve with a positive slope (first derivative) and a positive curvature (second derivative).  We could not infer the functional dependence of flow rate on diameter.  A log plot of f vs. d would yield a slope of 4 and an intercept equal to h/l.  This log plot shows us that the flow rate increases with diameter to the fourth power and that it is proportional to the incline of the tube (h/l).  

To produce a log plot, we could take the log of our data and plot that data on a standard scale, or we can plot the data on log graph paper where the paper takes the log.  The latter is easier, especially when there are many data points.

We can prove that a straight line on a plot of log p vs. log q yields:  

p (q) = kq

Taking the log on both sides of p =kq results in    

 log p = log k + n log q.   

This is the form of p = nq +k, where n is the slope and k is the intercept.  
The slope for any log p vs. log q plot is calculated from two points on the line. 

 The slope = log (p/p) / log (q/q).  

The intercept k occurs when q =1 because anything raised to the power or log of zero is one.  We can find directly on the graph the intercept if a q = 1 data point is on the line by reading the value of p when q =1.  If q =1 is not on the plot, we can calculate k from a data point on the line by the equation:

k = p/q