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Exponential Identities

For all real , m, and n, we have the following identities:

(1)  

(2)  

(3)    

(4)  

(5)    

(6)    

For all n and , the function is monotonically increasing in n.

When convenient, we shall assume .

Rates of Growth of Polynomials and Exponentials

For all real constants a and b such that a > 1

or

So any positive exponential function grows faster than any polynomial.

Using e = 2.71828

For all real x,

(1)  

(2)   where equality only holds when x = 0

When , we have the following approximation

(3)  

When , the approximation of by 1 + x can be stated as.

(4)   ,

Note that asymptotic notation is used to describe the limiting behavior as versus

Lastly, for all x,

(5)

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