Exponential Function Reference
This is the Exponential Function:
f(x) = a^{x}
a is any value greater than 0
Properties depend on value of "a"
 When a=1, the graph is a horizontal line at y=1
 Apart from that there are two cases to look at:
a between 0 and 1 
a above 1 

Example: f(x) = (0.5)^{x} 
Example: f(x) = (2)^{x} 

For a between 0 and 1

For a above 1:

Plot the graph here (use the "a" slider)
In General:
 It is always greater than 0, and never crosses the xaxis
 It always intersects the yaxis at y=1 ... in other words it passes through (0,1)
 At x=1, f(x)=a ... in other words it passes through (1,a)
 It is an Injective (onetoone) function
Its Domain is the Real Numbers:
Its Range is the Positive Real Numbers: (0, +∞)
Inverse
a^{x} is the inverse function of log_{a}(x) (the Logarithmic Function)
So the Exponential Function can be "reversed" by the Logarithmic Function.
The Natural Exponential Function
This is the "Natural" Exponential Function:
f(x) = e^{x}
Where e is "Eulers Number" = 2.718281828459 (and more ...)
Graph of f(x) = e^{x}
At the point (1,e) the slope of the line is e and the line is tangent to the curve.