**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.

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

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**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.

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

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**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 ^{( 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}

**Ratio Of Common To Natural Exponential**

Let *x* be a real number. The ratio (quotient) of the common exponential of *x* to the natural exponential of *x* is equal to the exponential of *x* to the base 10/ *e* :

10 ^{x} / *e *^{x} = (10/ *e* ) ^{x} ≈ (3.6788) ^{x}

Now suppose that *x* is some nonzero real number. The ratio (quotient) of the common exponential of 1/ *x* to the natural exponential 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} ≈ (3.6788) ^{1/ x}

**Ratio Of Natural To Common Exponential**

Let *x* be a real number. The ratio (quotient) of the natural exponential of *x* to the common exponential of *x* is equal to the exponential of *x* to the base *e* /10. That is to say:

e ^{x} /10 ^{x} = ( *e* /10) ^{x} ≈ (0.271828) ^{x}

Now suppose that *x* is some nonzero real number. The ratio (quotient) of the natural exponential of 1/ *x* to the common exponential of 1/ *x* is equal to the exponential of 1/ *x* to the base *e* /10:

( *e* ^{1/ x} )/(10 ^{1/ x} ) = ( *e* /10) ^{1/ x} ≈ (0.271828) ^{1/ x}

**Common Exponential Of Ratio**

Let *x* and *y* be real numbers, with the restriction that *y* ≠ 0. The common exponential of the ratio (quotient) of *x* to *y* is equal to the exponential of 1/ *y* to the base 10 ^{x} :

10 ^{x/y} = (10 ^{x} ) ^{1/ y}

A similar situation exists for base *e* . The natural exponential of the ratio (quotient) of *x* to *y* is equal to the exponential of 1/ *y* to the base *e *^{x} :

e ^{x/y} = ( *e *^{x} ) ^{1/ y}

Practice problems for these concepts can be found at: Logarithms, Exponentials, And Trigonometry for Physics Practice Test

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From Physics Demystified: A Self-Teaching Guide. Copyright © 2002 by The McGraw-Hill Companies, Inc. All Rights Reserved.