There are many situations where the increase or decrease of some variable in a fixed time interval will be proportional to the magnitude of the variable at the beginning of that time interval.
For example, let's look at a population of wee beasties which increases by 10% per year. If there were 100 wee beasties now, there would be an increase of 10 wee beasties after a year. We would see an increase of 500 wee beasties in a year when there were 5000 at the beginning.
Likewise, we can look at a population which decreases by 50% (i.e. a decrease to 1/2, or by a factor of 2) every day. A population of 100 would be down to 50 a day later, and a population of 5000 would drop to 2500 after one day.
These are all examples of exponential growth and decay.
A single equation can be used to solve all problems involving this type of change:
We can re-write this equation in another convenient form. Dividing the equation by , and then taking the natural logarithms of both sides, we get
These two problems are used to solve questions for both exponential growth and exponential decay.