import math import numpy as np
What is e? It is simply a number (known as Euler's number):
e is a significant number, because it is the base rate of growth shared by all continually growing processes.
For example, if I have 10 dollars, and it grows 100% in 1 year (compounding continuously), I end up with 10*e^1 dollars:
# 100% growth for 1 year 10 * np.exp(1)
# 100% growth for 2 years 10 * np.exp(2)
Side note: When e is raised to a power, it is known as the exponential function. Technically, any number can be the base, and it would still be known as an exponential function (such as 2^5). But in our context, the base of the exponential function is assumed to be e.
Anyway, what if I only have 20% growth instead of 100% growth?
# 20% growth for 1 year 10 * np.exp(0.20)
# 20% growth for 2 years 10 * np.exp(0.20 * 2)
What is the (natural) logarithm? It gives you the time needed to reach a certain level of growth. For example, if I want growth by a factor of 2.718, it will take me 1 unit of time (assuming a 100% growth rate):
# time needed to grow 1 unit to 2.718 units np.log(2.718)
If I want growth by a factor of 7.389, it will take me 2 units of time:
# time needed to grow 1 unit to 7.389 units np.log(7.389)
If I want growth by a factor of 1, it will take me 0 units of time:
# time needed to grow 1 unit to 1 unit np.log(1)
If I want growth by a factor of 0.5, it will take me -0.693 units of time (which is like looking back in time):
# time needed to grow 1 unit to 0.5 units np.log(0.5)
As you can see, the exponential function and the natural logarithm are inverses of one another: