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Exponential Functions for Physics Help

By — McGraw-Hill Professional
Updated on Sep 17, 2011

Introduction

An exponential is a number that results from the raising of a constant to a given power. Suppose that the following relationship exists among three real numbers a, x , and y :

a x = y

Then y is the base-a exponential of x . The two most common exponential-function bases are a = 10 and a = e ≈ 2.71828.

Common Exponentials

Base-10 exponentials are also known as common exponentials . For example:

10 −3.000 = 0.001

Figure 5-5 is an approximate linear-coordinate graph of the function y = 10 x . Figure 5-6 is the same graph in semilog coordinates. The domain encompasses the entire set of real numbers. The range is limited to the positive real numbers.

Logarithms, Exponentials, and Trigonometry Exponential Functions Reciprocal Of Common Exponential

Fig. 5-5 Approximate linear-coordinate graph of the common exponential function.

 

Logarithms, Exponentials, and Trigonometry Exponential Functions Product Of Exponentials

Fig. 5-6 Approximate semilog graph of the common exponential function.

Natural Exponentials

Base- e exponentials are also known as natural exponentials . For example:

e −3.000 ≈ 2.71828 −3.000 ≈ 0.04979

Figure 5-7 is an approximate linear-coordinate graph of the function y = e x . Figure 5-8 is the same graph in semilog coordinates. The domain encompasses the entire set of real numbers. The range is limited to the positive real numbers.

Logarithms, Exponentials, and Trigonometry Exponential Functions Exponential Of Common Exponential

Fig. 5-7 Approximate linear-coordinate graph of the natural exponential function.

 

Logarithms, Exponentials, and Trigonometry Exponential Functions Product Of Common And Natural Exponentials

Fig. 5-8 Approximate semilog graph of the natural exponential function.

Reciprocal Of Common Exponential

Let x be a real number. The reciprocal of the common exponential of x is equal to the common exponential of the additive inverse of x :

1/(10 x ) = 10 x

Reciprocal Of Natural Exponential

Let x be a real number. The reciprocal of the natural exponential of x is equal to the natural exponential of the additive inverse of x :

l/( e x ) = e −x

Product Of Exponentials

Let x and y be real numbers. The product of the exponentials of x and y is equal to the exponential of the sum of x and y . Both these equations hold true:

(10 x )(10 y ) = 10 ( x + y )

( e x )( e y ) = e ( x + y )

Ratio Of Exponentials

Let x and y be real numbers. The ratio (quotient) of the exponentials of x and y is equal to the exponential of the difference between x and y . Both these equations hold true:

10 x /10 y = 10 ( xy )

e x /e y = e ( xy )

Exponential Of Common Exponential

Suppose that x and y are real numbers. The y th power of the quantity 10 x is equal to the common exponential of the product xy :

(10 x ) y = 10 ( xy )

The same situation holds for base e . The y th power of the quantity e x is equal to the natural exponential of the product xy :

( e x ) y = e ( xy )

Product Of Common And Natural Exponentials

Let x be a real number. The product of the common and natural exponentials of x is equal to the exponential of x to the base 10 e . That is to say:

(10 x )( e x ) = (10 e ) x ≈ (27.1828) x

Now suppose that x is some nonzero real number. The product of the common and natural exponentials of 1/ x is equal to the exponential of 1/ x to the base 10 e :

(10 1/ x )( e 1/ x ) = (10 e ) 1 x ≈ (27.1828) 1/ x

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