## 4. Exponential functions from the ground up

Mathematically we start with the growth of the population (P) over time (t) such that:

dP/dt = kP

Over a time interval, denoted by Δt ('delta-t'), the mathematical solution to this equation is:

P_{(t+Δt)} = P_{t}. exp(k.Δt)

where P_{t} and P_{(t+Δt)} are the population sizes at times t and (t + Δt) respectively.

This is an * exponential function*, hence the name

*. The expression:*

**exponential growth**exp(k.Δt) is another way of writing e^{(k.Δt)}

[Technical note: e is called Euler's number or Napier's constant and is one of the most important numbers in mathematics. The other important numbers are 1, 0, the imaginary value i and π. e is a constant, with a value of 2.71828 18284 59045 23536... In mathematical terms e is the unique real number so that the derivative of the function f(x) = e^{x} at the point where x = 0 is equal to 1. The function e^{x} is called the exponential function and the inverse of this number is the natural or Naperian logarithm. Find out more at: