Properties of Exponential Functions

    Exponential and logarithmic functions are used throughout the sciences and are particularly useful in business applications. For example, most banks in the world use the exponential function with compound interest. In this topic we concentrate on decay and growth models, which are two different types of exponential functions, and we emphasize graphs and the properties of logarithms. We also show by example how to solve exponential and logarithmic equations.

    Recall that is just multiplied together times; and that and provided Also, if and are integers with in reduced form, then Thus, we know what where is a rational number means. To extend for a real number it is customary to use the following completeness axiom.

Definition (Completeness Axiom) For any real number there exist rational numbers such that which means for any there exists a number such that whenever

Definition (Exponential Function) If is a real number with and then the function is an exponential function with base and is defined by where is a sequence of rational numbers such that

Example (Exponential Function) The number has infinite decimal representation which means that can be approximated to any desired degree of accuracy by members of the sequence of rational numbers Therefore, the exponential number is given by the limit of the sequence of numbers

Proposition (Properties of Exponentials) Let and be real numbers, and positive real numbers.

    (i) If then if and only if
    
    (ii) If and then
    
    (iii) If and then
    
    (iv)
    
    (v)
    
    (vi) and
    

Example (Evaluating Exponential Functions) Use a scientific calculator to compute and

Example (Graphing Exponential Functions) Sketch the following graphs.

(a) Plot the functions by either using a graphing calculator or if you can use transformations given

    Solution. To graph we can graph and apply a horizontal shift left 2.  



(b)  Plot the functions by either using a graphing calculator or if you can use transformations given .

    Solution. To graph we can graph and reflect through the -axis.



(c) Plot the functions by either using a graphing calculator or if you can use transformations given .

    Solution. To graph we can graph and scale by a factor of 2.

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Cite this as:
Properties Of Exponential Functions
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
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