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Line 22: − − ==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. − − The derivative of exponential functions, − − <math>\frac{d}{dx}a^{x}=a^{x}\ln{a}</math> − − 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.
exponential functions moved to their own page
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 · 2 · 2 · 2 · 2 · 2 = 64
*2<sup>6</sup>, "two to the sixth power," 2 · 2 · 2 · 2 · 2 · 2 = 64
*''x''<sup>4</sup>, "''x'' to the fourth power," ''x'' · ''x'' · ''x'' · ''x''
*''x''<sup>4</sup>, "''x'' to the fourth power," ''x'' · ''x'' · ''x'' · ''x''
*''y''<sup>''n''</sup>, "''y'' to the n-th power," ''y'' · ''y'' · ''y'' · ... · ''y'', where ''y'' appears ''n'' times.
*''y''<sup>''n''</sup>, "''y'' to the n-th power," ''y'' · ''y'' · ''y'' · ... · ''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—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
The definition of exponentiation as repeated multiplication only makes sense when the exponent is a positive integer—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>) · (''x''<sup>b</sup>) = ''x''<sup>(a + b)</sup>
*(''x''<sup>a</sup>) · (''x''<sup>b</sup>) = ''x''<sup>(a + b)</sup>
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>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]]
[[Category:Mathematics]]