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exponentials

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Definitions of exponentials in various dictionaries:

noun - a function in which an independent variable appears as an exponent

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Possible Dictionary Clues
Plural form of exponential.
(with reference to an increase) more and more rapidly.
By means of or as expressed by a mathematical exponent.

Exponentials might refer to
In mathematics, an Exponential function is a function of the form* in which the argument x occurs as an exponent. A function of the form * * * * f * ( * x * ) * = * * b * * x * + * c * * * , * * * {\displaystyle f(x)=b^{x+c},} * where c is a constant, is also considered an exponential function and can be rewritten as * * * * f * ( * x * ) * = * a * * b * * x * * * , * * * {\displaystyle f(x)=ab^{x},} * with * * * * a * = * * b * * c * * * . * * * {\displaystyle a=b^{c}.} * * As functions of a real variable, exponential functions are uniquely characterized by the fact that the growth rate of such a function (that is, its derivative) is directly proportional to the value of the function. The constant of proportionality of this relationship is the natural logarithm of the base b: * * The constant e = 2.71828... is the unique base for which the constant of proportionality is 1, so that the function's derivative is itself: * * Since changing the base of the exponential function merely results in the appearance of an additional constant factor, it is computationally convenient to reduce the study of exponential functions in mathematical analysis to the study of this particular function, conventionally called the "natural exponential function", or simply, "the exponential function" and denoted by * * While both notations are common, the former notation is generally used for simpler exponents, while the latter tends to be used when the exponent is a complicated expression. * The exponential function satisfies the fundamental multiplicative identity * * This identity extends to complex-valued exponents. It can be shown that every continuous, nonzero solution of the functional equation * * * * f * ( * x * + * y * ) * = * f * ( * x * ) * f * ( * y * ) * * * {\displaystyle f(x+y)=f(x)f(y)} * is an exponential function, * * * * f * : * * R * * → * * R * * , * * x * ↦ * * b * * x * * * , * * * {\displaystyle f:\mathbb {R} \to \mathbb {R} ,\ x\mapsto b^{x},} * with * * * * b * > * 0. * * * {\displaystyle b>0.} * The fundamental multiplicative identity, along with the definition of the number e as e1, shows that * * * * * e * * n * * * = * * * * * e * × * ⋯ * × * ...

Exponentials might refer to

In mathematics, an Exponential function is a function of the form* in which the argument x occurs as an exponent. A function of the form
*
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* =
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* b
*
* x
* +
* c
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* ,
*
*
* {\displaystyle f(x)=b^{x+c},}
* where c is a constant, is also considered an exponential function and can be rewritten as
*
*
*
* f
* (
* x
* )
* =
* a
*
* b
*
* x
*
*
* ,
*
*
* {\displaystyle f(x)=ab^{x},}
* with
*
*
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* a
* =
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* c
*
*
* .
*
*
* {\displaystyle a=b^{c}.}
*
* As functions of a real variable, exponential functions are uniquely characterized by the fact that the growth rate of such a function (that is, its derivative) is directly proportional to the value of the function. The constant of proportionality of this relationship is the natural logarithm of the base b:
*
* The constant e = 2.71828... is the unique base for which the constant of proportionality is 1, so that the function's derivative is itself:
*
* Since changing the base of the exponential function merely results in the appearance of an additional constant factor, it is computationally convenient to reduce the study of exponential functions in mathematical analysis to the study of this particular function, conventionally called the "natural exponential function", or simply, "the exponential function" and denoted by
*
* While both notations are common, the former notation is generally used for simpler exponents, while the latter tends to be used when the exponent is a complicated expression.
* The exponential function satisfies the fundamental multiplicative identity
*
* This identity extends to complex-valued exponents. It can be shown that every continuous, nonzero solution of the functional equation
*
*
*
* f
* (
* x
* +
* y
* )
* =
* f
* (
* x
* )
* f
* (
* y
* )
*
*
* {\displaystyle f(x+y)=f(x)f(y)}
* is an exponential function,
*
*
*
* f
* :
*
* R
*
* →
*
* R
*
* ,
*
* x
* ↦
*
* b
*
* x
*
*
* ,
*
*
* {\displaystyle f:\mathbb {R} \to \mathbb {R} ,\ x\mapsto b^{x},}
* with
*
*
*
* b
* >
* 0.
*
*
* {\displaystyle b>0.}
* The fundamental multiplicative identity, along with the definition of the number e as e1, shows that
*
*
*
*
* e
*
* n
*
*
* =
*
*
*
*
* e
* ×
* ⋯
* ×
* ...

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