Exponential Growth

A quantity is said to grow exponentially when, the rate of growth of the quantity is directly proportional to the quantity’s present value. The exactly opposite of exponential growth is exponential decay. In exponential decay, the quantity decreases. In exponential growth the quantity increases. If the growing quantity can assume only discrete values, then this exponential-growth is also called geometric growth. It is similar to the terms of a geometric sequence. Another name for exponential growth activity is Malthusian growth model.

 

The formula for finding the value of a given variable x, given some exponential growth data, that is growing with the rate of growth equal to r with respect to time t, where time is treated as a discrete variable (that means the values of t can be 0,1,2,3…. etc and not decimals or fractions), is as follows:

X(t) = Xo * (1+r)^t

Here the rate r is taken in absolute terms and not as a percent. Xo is the initial value of the quantity. That means, Xo is the value of the quantity when time t = 0. For example, let us now try to make an exponential growth chart for some quantity x, that is increasing at the rate of 2% per year. Therefore our time t here would represent the number of years, that is, 0,1,2,…. Also, we are given that at time t = 0, the initial value of the quantity X is Xo which is equal to 1000. Then the values on the chart would look like this:

 

 

X(t) = Xo * (1+r)^t = 1000 * (1+0.02)^t = 1000 * (1.02)^t.

(Note that we took r = 0.02 since 2% in absolute terms would equal to 2/100 = 0.02)

t

X(t) = 1000 * (1.02)^t

0

1000

1

1020

2

1040.4

3

1061.208

4

1082.432

5

1104.081

6

1126.162

7

1148.686

8

1171.659

9

1195.093

10

1218.994

 

Now suppose the above chart represents the population of a certain remote village. Then the exponential population growth graph of that village can be plotted using the above data. We would take the time in number of years on the x axis and the population on the y axis. The graph would look as follows: