Understanding e and Plotting Curves on Log Scales

We can see that the solution to the previous differential equation brought up the exponential constant V, where e 2.718. We can ask — why do circuits like this always seem to lead 

But this in turn can be traced back to the observation that the exponential constant e itself happens to be one of the most natural parameters of our world. The following example illustrates this. 

Note that the simplest and most obvious initial assumption of a constant failure rate has actually led to an exponential curve. That is because the exponential curve is simply a succession of evenly spaced data points (very close to each other), that are in simple geometric progression, that is, the ratio of any point to its preceding point is a constant. Most natural processes behave similarly, and so e is encountered very frequently. 

If we make the vertical scale (only) logarithmic rather than linear, we will find this gives us a straight line. Why so That actually comes about due to a useful property of the logarithm as described next. 

The definition of a logarithm is as follows — if A = Bc, then logB(A) = log(C), where logs (A) is the base-B logarithm of A. The commonly referred to logarithm, or just log, has an implied base of 10, whereas the natural logarithm In is an abbreviation for a base-e logarithm. 

---