Exponential Functions
Laws of Exponents
If s, t, a and b are real numbers, with a > 0 and b > 0, then:
1) a^{s} Â· a^{t} = a^{s + t}
2) (a^{s})^{t} = a^{st}
3) (ab)^{s} = a^{s} Â· b^{s}
4)
5)
6)
7) 1^{s} = 0
8) a^{0} = 1
Exponential Functions
Definition: If b is a positive real number, b ≠ 1, the exponential function with base b is f(x)
= b^{x} , for every
real number x.
Examples:
f(x) = 2^{x} 
f(x) = 3^{x} 






Properties of Exponential Functions of the Form f(x) = b^{x}
1. b^{x} > 0 for all numbers, x; so, the range of these functions is (0,
∞).
2. Their domains will be (∞,
∞).
3. All will have yintercepts at (0,1) and no xintercepts.
4. The xaxis is a horizontal asymptote.
5. If b > 1, f(x) = b^{x} is an increasing function. If 0 < b < 1, f(x)
= b^{x} is a decreasing function.
6. They are all onetoone.
Here is a graph with f(x) = 2^{x} , f(x) = 3^{x} and f(x) = 4^{x} all on the same screen. Can you tell which is which?
The Natural Exponential Function: f(x) = e^{x}
e is a special number in mathematics, and is approximately 2.7182818. Since e
> 1, the graph of the exponential function is shaped like the graph of f(x) = 2^{x}. It is an increasing function. e^{x}
> 0 for all numbers x. Here is the graph of f(x) = e^{x}:
Exponential Growth Functions Any function of the form
y = a(b^{x}), b > 1, a > 0, is an exponential growth function. Its graph will increase on its entire domain.

Exponential Decay Function Any function of the form y
= a(b^{x}), b > 1, a > 0, or y = a(c ), 0 c 1, a 0, is an exponential decay function, and will decrease on its domain.



