Exponentiation - Revision history

GregG: rewrite the last paragraph

2012-07-01T00:42:52Z

rewrite the last paragraph

DavidE: expanded on last paragraph; added internal link; minor correction

2009-12-14T15:13:28Z

expanded on last paragraph; added internal link; minor correction

Foxtrot: exponential functions moved to their own page

2009-02-05T08:31:26Z

exponential functions moved to their own page

BRichtigen: /* Exponential functions */

2008-12-11T14:09:13Z

Exponential functions

DaBoss3: /* Exponential functions */ Made this more consistent

2008-04-25T01:01:57Z

Exponential functions: Made this more consistent

DaBoss3: /* Exponential functions */

2008-04-23T23:31:27Z

Exponential functions

DaBoss3 at 23:30, April 23, 2008

2008-04-23T23:30:15Z

Jallen at 06:31, July 27, 2007

2007-07-27T06:31:23Z

Pretzel: not all roots are surds; little bit about ''e''

2007-06-30T11:36:43Z

not all roots are surds; little bit about ''e''

Dpbsmith at 19:34, June 19, 2007

2007-06-19T19:34:34Z

@@ Line 21: / Line 21: @@
 *(''x''<sup>0.5</sup>) &middot; (''x''<sup>0.5</sup>) = ''x''<sup>1</sup> = ''x''
-The sequence x<sup>0</sup>, x<sup>1</sup>, x<sup>2</sup>, x<sup>3</sup>, ... is an example of what can be called a [[geometrical progression]], [[compound interest]] growth, or [[exponential growth]]. These are all different names for the same thing. Because of the compounding effect, exponential growth occurs faster and faster... literally like wildfire. This has led to the colloquial use of the word ''exponential'' to mean "growing very rapidly," or even "very large."  The same sequence except with negative exponents would describe [[exponential decay]].
+A geometric sequence can be used to represent [[compound interest]], [[exponential growth]], or [[exponential decay]].  Because of the compounding effect, exponential growth occurs faster and faster, like wildfire. This has led to the colloquial use of the word ''exponential'' to mean "growing very rapidly," or even "very large."
 [[Category:Mathematics]]

@@ Line 21: / Line 21: @@
 *(''x''<sup>0.5</sup>) &middot; (''x''<sup>0.5</sup>) = ''x''<sup>1</sup> = ''x''
-The sequence x<sup>1</sup>, x<sup>1</sup>, x<sup>2</sup>, x<sup>3</sup>, ... is an example of what can be called a [[geometrical progression]], [[compound interest]] growth, or [[exponential growth]]. These are all different names for the same thing. Because of the compounding effect, exponential growth occurs faster and faster... literally like wildfire. This has led to the colloquial use of the word ''exponential'' to mean "growing very rapidly," or even "very large."
+The sequence x<sup>0</sup>, x<sup>1</sup>, x<sup>2</sup>, x<sup>3</sup>, ... is an example of what can be called a [[geometrical progression]], [[compound interest]] growth, or [[exponential growth]]. These are all different names for the same thing. Because of the compounding effect, exponential growth occurs faster and faster... literally like wildfire. This has led to the colloquial use of the word ''exponential'' to mean "growing very rapidly," or even "very large."  The same sequence except with negative exponents would describe [[exponential decay]].
 [[Category:Mathematics]]

@@ Line 1: / Line 1: @@
-In mathematics, '''exponentiation''' is the name for the operation also called ''raising to a power.'' In simple cases, it refers to repeated multiplication. It is indicated by a ''superscript,'' a small number or expression written above the line:
+In mathematics, '''exponentiation''' is the name for the operation also called ''raising to a power.'' In simple cases, it refers to repeated [[multiplication]]. It is indicated by a ''superscript,'' a small number or expression written above the line:
 *2<sup>6</sup>, "two to the sixth power," 2 &middot; 2 &middot; 2 &middot; 2 &middot; 2 &middot; 2 = 64
 *''x''<sup>4</sup>, "''x'' to the fourth power," ''x'' &middot; ''x'' &middot; ''x'' &middot; ''x''
 *''y''<sup>''n''</sup>, "''y'' to the n-th power," ''y'' &middot; ''y'' &middot; ''y'' &middot; ... &middot; ''y'', where ''y'' appears ''n'' times.
-The superscripted value is called the ''exponent.''
+The superscripted value is called the [[exponent]]. The definition of exponentiation leads to the notion of [[exponential function]]s, where the exponent becomes a [[variable]].
 The definition of exponentiation as repeated multiplication only makes sense when the exponent is a positive integer&mdash;what does it mean to say "x multiplied by itself half a time" or "minus three times?" However, mathematicians have found logical meanings for zero, negative, fractional, and even complex exponents. These meanings arise from the basic observation that
 *(''x''<sup>a</sup>) &middot; (''x''<sup>b</sup>) = ''x''<sup>(a + b)</sup>
@@ Line 21: / Line 22: @@
 The sequence x<sup>1</sup>, x<sup>1</sup>, x<sup>2</sup>, x<sup>3</sup>, ... is an example of what can be called a [[geometrical progression]], [[compound interest]] growth, or [[exponential growth]]. These are all different names for the same thing. Because of the compounding effect, exponential growth occurs faster and faster... literally like wildfire. This has led to the colloquial use of the word ''exponential'' to mean "growing very rapidly," or even "very large."
 [[Category:Mathematics]]

@@ Line 23: / Line 23: @@
 ==Exponential functions==
+[[Image:Exp.png|right|pxl=200|thumb|exponential functions]]
 Functions of the form <math>f(s)=a^{s}</math>, where ''a'' is constant and ''s'' is a [[complex number]] are known as exponential functions.

@@ Line 28: / Line 28: @@
 The derivative of exponential functions,
-<math>\frac{d}{dx}a^{x}=s^{x}\ln{s}</math>
+<math>\frac{d}{dx}a^{x}=a^{x}\ln{a}</math>
-Where <math>\ln{s}</math> is the [[natural logarithm]] for ''s''.
+Where <math>\ln{a}</math> is the [[natural logarithm]] for ''a''.
 The [[transcendental number]] ''[[e]]'' (2.7182818...) has the property that the [[derivative]] of the function ''e''<sup>''x''</sup> is ''e''<sup>''x''</sup>.  This function is thus important in the solution of many types of differential equations.
 [[Category:Mathematics]]

← Older revision		Revision as of 06:31, July 27, 2007
Line 23:		Line 23:

	The [[transcendental number]] ''[[e]]'' (2.7182818...) has the property that the [[derivative]] of the function ''e''<sup>''x''</sup> is ''e''<sup>''x''</sup>. This function is thus important in the solution of many types of differential equations.		The [[transcendental number]] ''[[e]]'' (2.7182818...) has the property that the [[derivative]] of the function ''e''<sup>''x''</sup> is ''e''<sup>''x''</sup>. This function is thus important in the solution of many types of differential equations.
		+
		+	[[Category:Mathematics]]

@@ Line 17: / Line 17: @@
 *(''x''<sup>-n</sup>) = 1 / ''x''<sup>n</sup>
-Fractional exponents give us [[surd]]s; for example,  x<sup>0.5</sup> gives us the square root:
+Fractional exponents 1/''n'' give us the n-th root; for example,  x<sup>0.5</sup> gives us the square root:
 *(''x''<sup>0.5</sup>) &middot; (''x''<sup>0.5</sup>) = ''x''<sup>1</sup> = ''x''
 The sequence x<sup>1</sup>, x<sup>1</sup>, x<sup>2</sup>, x<sup>3</sup>, ... is an example of what can be called a [[geometrical progression]], [[compound interest]] growth, or [[exponential growth]]. These are all different names for the same thing. Because of the compounding effect, exponential growth occurs faster and faster... literally like wildfire. This has led to the colloquial use of the word ''exponential'' to mean "growing very rapidly," or even "very large."

← Older revision		Revision as of 19:34, June 19, 2007
Line 19:		Line 19:
	Fractional exponents give us [[surd]]s; for example, x<sup>0.5</sup> gives us the square root:		Fractional exponents give us [[surd]]s; for example, x<sup>0.5</sup> gives us the square root:
	*(''x''<sup>0.5</sup>) · (''x''<sup>0.5</sup>) = ''x''<sup>1</sup> = ''x''		*(''x''<sup>0.5</sup>) · (''x''<sup>0.5</sup>) = ''x''<sup>1</sup> = ''x''
		+
		+	The sequence x<sup>1</sup>, x<sup>1</sup>, x<sup>2</sup>, x<sup>3</sup>, ... is an example of what can be called a [[geometrical progression]], [[compound interest]] growth, or [[exponential growth]]. These are all different names for the same thing. Because of the compounding effect, exponential growth occurs faster and faster... literally like wildfire. This has led to the colloquial use of the word ''exponential'' to mean "growing very rapidly," or even "very large."