Consider a variable  to be some function , that is . If  increases from  to  then the change in the value of  will be given by . The proportional change, however, is given by:
The rate of proportional change at the point  is given by the above expression divided by the step change in the  value, namely :
Rate of proportional change 
Using calculus, we know that
therefore the rate of proportional change equals:
This quantity serves as a measure of the rate of proportional change of the function . Just as  measures the gradient of the curve  plotted on a linear scale,  measures the slope of the curve when plotted on a semi-logarithmic scale, that is the rate of proportional change. For example, a value of  means that the curve increases at  per unit .
The same argument can be applied to the case when we plot a function on both  and  logarithmic scales. In such a case, the following result is true: