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 basea exponential of x . The two most common exponentialfunction bases are a = 10 and a = e ≈ 2.71828.
Common Exponentials
Base10 exponentials are also known as common exponentials . For example:
10 ^{−3.000} = 0.001
Figure 55 is an approximate linearcoordinate graph of the function y = 10 ^{x} . Figure 56 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.
Fig. 55 Approximate linearcoordinate graph of the common exponential function.
Fig. 56 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 57 is an approximate linearcoordinate graph of the function y = e ^{x} . Figure 58 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.
Fig. 57 Approximate linearcoordinate graph of the natural exponential function.
Fig. 58 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 ^{( x − y )}
e ^{x} /e ^{y} = e ^{( x − y )}
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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