The situation where a quantity is multiplied by the same number every time step is
called exponential growth. This number is called the growth rate for that time step. If
is the growth rate then:
To determine whether a quantity is growing exponentially, you can determine the ratios of consecutive values of the quantity. If these ratios are approximately equal, you can say that the growth is exponential. The amount at is called the initial value.
If a quantity increases with the same percentage each time step we say it grows exponentially.
If the growth is
percent per time step, the growth rate is:
For the quantity increases and : exponential growth.
For the quantity decreases and : exponential decay.
When working with exponential growth you use powers: when you multiply
times with the same number
, you write
. This is a power, the growth rate
is called the base,
is called the exponent, where
(for now) is a positive integer.
For we agree that .
For any base and any positive integers and the following rules apply: