# Exponential growth

Exponential growth refers to a manner in which a quantity increases when its rate of growth is proportional to itself. For example, a quantity that keeps tripling is an example of a quantity that exhibits exponential growth; since the value is constantly tripling with respect to itself, the rate of increase is also constantly increasing. The growth of bacterial colonies is an example of something in nature that can exhibit exponential growth. Exponential growth can be modeled using the following function:

x(t) = ae^{kt}

where x(t) is the value of the function at time, t, a is the initial value, and k is the growth rate.

Example

If you start with 4 bacteria, and 20 minutes later there are 8, assuming that the bacteria continue multiplying at the same rate, how many bacteria will there be in 5 hours?

First solve for k. a = 4 is the initial number of bacteria we started with. t is 20 minutes, and x(20) = 8. Plugging this all into the formula, we can solve for k:

8 = 4 × e^{20k}

2 = e^{20k}

ln2 = ln(e^{20k}) = 20k

k = ln2 / 20

Now we can substitute this value for k into the equation along with a new time of 5 hours, or 300 minutes, to find out how many bacteria we will have after 5 hours.

x(300) = 4 × e^{300(ln2 / 20)} = 131,072

In only 5 hours, the 4 bacteria we started with will multiply to a colony with 131,072 bacteria, given that they continue multiplying at the same rate.

## Exponential decay

Exponential decay is more or less the same as exponential growth, except that the exponential function decreases over time rather than increases. The same formula,

x(t) = ae^{kt}

can be used in the same manner as the example above. The only difference is that the growth factor, k, is a negative value. This will cause the function to decrease rather than increase with increasing t.