With exponential growth you have to multiply by the same number every unit of time.
This number is called the growth rate for that time unit. If
is the growth rate then:
To be able to use negative and/or rational exponents we need to agree about the following:
These are valid for and positive integer .
Both completely fit in with the computational rules for powers, such as:
This shows that a power for is meaningful if the exponent is either a positive number, a zero, a negative or a rational number.
can be any real number in fact.
And that is why the graph of an exponential function can be drawn as smooth curves.
Here you see the graph of .