[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/all2en\/wiki32\/basic-expounts-a-wikipedia\/#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/all2en\/wiki32\/basic-expounts-a-wikipedia\/","headline":"Basic expounts A \u2014 Wikipedia","name":"Basic expounts A \u2014 Wikipedia","description":"before-content-x4 In real analysis, the basic exponential a is the noted function exp a who, to all real x ,","datePublished":"2021-05-28","dateModified":"2021-05-28","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/all2en\/wiki32\/author\/lordneo\/#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/all2en\/wiki32\/author\/lordneo\/","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/44a4cee54c4c053e967fe3e7d054edd4?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/44a4cee54c4c053e967fe3e7d054edd4?s=96&d=mm&r=g","height":96,"width":96}},"publisher":{"@type":"Organization","name":"Enzyklop\u00e4die","logo":{"@type":"ImageObject","@id":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","url":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","width":600,"height":60}},"image":{"@type":"ImageObject","@id":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/181a7f23dd4f4aa91180b51a3a5f5574f68313e8","url":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/181a7f23dd4f4aa91180b51a3a5f5574f68313e8","height":"","width":""},"url":"https:\/\/wiki.edu.vn\/all2en\/wiki32\/basic-expounts-a-wikipedia\/","wordCount":8517,"articleBody":"     (adsbygoogle = window.adsbygoogle || []).push({});before-content-x4In real analysis, the basic exponential a is the noted function exp a who, to all real x , associates the real a x . She only makes sense for a real a strictly positive. It extends to all the reals the function, defined on all natural integers, which to the whole n associated a n . It is therefore the continuous version of a geometric suite.        (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4It is expressed using the exponential usual functions and the nepalian logarithm in the form exp a\u2061 ( x ) = a x= exln\u2061(a). {displaystyle exp _{a}(x)=a^{x}={rm {e}}^{xln(a)}.} It can be defined as the only continuous function on \u211d, taking the value a in first and transforming a sum into a product.        (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4For a different from first , this is the reciprocal of the basic logarithm function a . These functions are sometimes called anti -data functions. The case a = and corresponds to the exponential functions and Nep\u00e9rian logarithm. Exponential functions are the only derivable functions on \u211d, proportional to their derivative and taking the value first in 0 . They make it possible to model the physical or biological phenomena in which the speed of growth is proportional to the size of the population. We also find the term exponential functions for functions whose expression is N a x .          (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4We consider a real a strictly positive; It is easy to define a n like the product of a by himself n times for any whole n greater than or equal to first , exp a\u2061 ( n ) = a n= a\u00d7a\u00d7\u22ef\u00d7a\u23dfn\u00a0fois{displaystyle exp _{a}(n)=a^{n}={underset {n{text{ fois}}}{underbrace {atimes atimes cdots times a} }}} then define a 0 = 1 And a \u2013n = first \/ a n . We easily demonstrate the property a n + m = a n \u00d7 a m . This fairly natural construction corresponds to the so -called growth or exponential decrease phenomena. Example 1: Imagine a population whose size increases by 30% every 10 years. If we note N the population in 1900, it is easy to calculate the population in 1910, 1920 … which will be N \u00d7 1.3 , Then N \u00d7 1.3 2 \u2026 To reach the end of n decades N \u00d7 1.3 n . It is even possible to determine the population in 1890, 1880 … which will be N \u00d7 1.3 \u22121 , N \u00d7 1.3 -2 … Example 2: Carbon 14 has a radioactive period of period T = 5 730 years which means that all T years, the number of radioactive particles has been divided by 2. If we measure, at a given moment, the number N radioactive particles, after n periods, the number of radioactive particles is only N \u00d7 (1\/2) n . The question that arises is to determine the size of the population or the number of radioactive particles between two measures (the decade for the population or the period for the particle). It is therefore a question of “filling the holes between the whole”. An attempt can be made thanks to the root n -th: if the population has multiplied in 10 years by 1.3, we are trying to determine by how multiplied it is each year. It is multiplied by a real q such as q ten = 1.3 , that’s to say q = ten \u221a 1.3 that we note 1.3 1\/10 . We are therefore able to define a r For non -whole exhibitors: exp a\u2061 ( first \/ q ) = a 1\/q= aq{displaystyle exp _{a}(1\/q)=a^{1\/q}={sqrt[{q}]{a}}} exp a\u2061 ( p \/ q ) = a p\/q= ( aq) p= apq{displaystyle exp _{a}(p\/q)=a^{p\/q}=({sqrt[{q}]{a}})^{p}={sqrt[{q}]{a^{p}}}} . We have “filled the holes” and defined a r for everything r rational. To define a x for all real x , we must add an argument of continuity: all real x is “as close as we want” of a rational p \/ q ; the value of a x will then be “close to” a p \/ q . This intuitive idea of \u200b\u200bwhat could be a x appears very early-at the same time as the exponential notation, that is to say from the XVII It is century [ first ] . But it will be necessary to wait for the following centuries to see in x \u21a6 a x : a function; verifying a x + y = a x a and , that is to say transforming a sum into a product; continue\u00a0; reciprocal of a logarithm function (which transforms a product in short); Derivable and whose derivative is proportional to the function. There are several possible entry points for the definition of exponential function: by its algebraic properties (transforms a sum into a product), by the property of its derivative (derivative proportional to the function), or by its relations with the exponential function and the Neper Logarithm function. Table of ContentsBy algebraic property [ modifier | Modifier and code ] Using the exponential function and the Neper Logarithm function [ modifier | Modifier and code ] By a differential equation [ modifier | Modifier and code ] As a reciprocal of the logarithm functions [ modifier | Modifier and code ] Algebraic properties [ modifier | Modifier and code ] Function study [ modifier | Modifier and code ] By algebraic property [ modifier | Modifier and code ] Definition – We call real exponential function, any function of R In R , not identically zero and continuous in at least one point, transforming a sum into a product, that is to say checking the functional equation  \u2200 in , in \u2208 R f ( in + in ) = f ( in ) \u00d7 f ( in ) . {displaystyle forall u,vin mathbb {R} quad f(u+v)=f(u)times f(v).}  Such a function f is continuous and strictly positive and for any real a > 0 , the special one f such as f (1) = a is called basic exponential a and is noted exp a . In other words: these functions are the continuous morphisms of ( R , +) in ( R + *, \u00d7), and are in bijection with R + * via  f \u21a6 f (first) . The relationship f ( in ) = f ( 2u2) = [f(u2)]2{displaystyle f(u)=fleft(2{frac {u}{2}}right)=left[fleft({frac {u}{2}}right)right]^{2}} ensures that the function is with positive values. The functional equation also guarantees that all these values \u200b\u200bare not zero as soon as one of them is. Then, considerations similar to those developed in the previous section ensure the existence [ 2 ] and uniqueness [ 3 ] , for any real a > 0 , of a function f defined on rational , checking the functional equation, and taking in 1 the value a . We demonstrate [ 3 ] continuity and – by density of \u211a in \u211d – The uniqueness of a function verifying the functional equation, taking in first the value a , and continues in at least one point. Its existence is obtained by extension by continuity: Details It is easy to check that the function x \u21a6 a x (defined for the moment only on rational) is monotonous and that the following a 1\/2 n Converges to 1. This, attached to the functional equation, makes it possible to show that the function is Cauchy-Continue on \u211a, therefore extendable by continuity to \u211d. By continuity and density, this extension to \u211d still verifies the functional equation. We can notice that – apart from the constant function first , that corresponds to a = 1 – All these applications f \u00a0: \u211d \u2192 ]0, +\u221e[ are bijective. So these are isomorphisms of ( R , +) in ( R + *, \u00d7). We prove that then f is differential and checks the differential equation:  f \u2032 ( x ) = f \u2032 ( 0 ) \u00d7 f ( x ) And f ( 0 ) = first. {displaystyle f'(x)=f'(0)times f(x)quad {text{et}}quad f(0)=1.}  Demonstrations first re method. -According to \u00a7 “by a differential equation” below, the functions g k : x \u21a6 exp( kx ) check g’ k = kg k , g k (0) = 1 And transform the sums into products. For k = exp \u22121 ( a ) , on a g k (1) = a therefore by uniqueness, g k = f . 2 It is method. – To demonstrate that a continuous function transforming a sum into a product is necessarily derivable, we can rely on the fact that a continuous function has primitives [ 4 ] . If we note F a primitive of f , we can write \u222b01f(x)f(t)dt=f(x)\u222b01f(t)dt=f(x)(F(1)\u2212F(0)){displaystyle int _{0}^{1}f(x)f(t)mathrm {d} t=f(x)int _{0}^{1}f(t)mathrm {d} t=f(x)(F(1)-F(0))} but also \u222b01f(x)f(t)dt=\u222b01f(x+t)dt=F(x+1)\u2212F(x).{displaystyle int _{0}^{1}f(x)f(t)mathrm {d} t=int _{0}^{1}f(x+t)mathrm {d} t=F(x+1)-F(x).} Function f being strictly positive, F is strictly increasing and F (first) – F (0) is then not zero. By confronting the two equalities, we can write f(x)=F(x+1)\u2212F(x)F(1)\u2212F(0),{displaystyle f(x)={frac {F(x+1)-F(x)}{F(1)-F(0)}},} which proves that, f speaking as a linear combination of derivable functions, f is differentiable. By deriving equality f(x+u)=f(x)f(u){displaystyle f(x+u)=f(x)f(u)} compared to x , we obtain f\u2032(x+u)=f\u2032(x)f(u){displaystyle f'(x+u)=f'(x)f(u)} Then, taking x equal to 0 , f\u2032(u)=f\u2032(0)f(u).{displaystyle f'(u)=f'(0)f(u).} Using the exponential function and the Neper Logarithm function [ modifier | Modifier and code ] Definition – Either a A strictly positive real. We call basic exponential function a the function defined on \u211d by  f ( x ) = exln\u2061(a){displaystyle f(x)={rm {e}}^{xln(a)},}  Or x \u21a6 and x is the exponential function and ln The Neper Logarithm function. This function is very continuous, transforms a sum into a product and takes the value a in 1. By a differential equation [ modifier | Modifier and code ] Definition – An exponential function is called any derivable function verifying the differential equation and the following initial condition:  f \u2032 = k f And f ( 0 ) = first {displaystyle f’=kfquad {text{et}}quad f(0)=1}  Or k is real. We can notice that for such a function, k is the value of the derivative in 0. Assuming only known the existence of a solution to k = 1 (function exp ), an obvious solution for k any function x \u21a6 exp( kx ) . We show [ 5 ] that this solution is the only one. In addition, the solution transforms any amount into a product [ 6 ] , therefore its definition coincides with that above “by algebraic property”, to a = exp( k ) . As a reciprocal of the logarithm functions [ modifier | Modifier and code ] Definition – Either a A strictly positive real, different from first . The basic logarithm function a is a bijection of R* + In R . We call basic exponential function a His reciprocal bijection:  (x \u2208 R , and = exp a\u2061 ( x ) )\u21d4 (and \u2208 ] 0,+\u221e[ , x = log a\u2061 ( and ) ). {displaystyle {bigl (}xin mathbb {R} ,quad y=exp _{a}(x){bigr )}Leftrightarrow {bigl (}yin left]0,+infty right[,quad x=log _{a}(y){bigr )}.}  The logarithm function being continuous, transforming a product in short and taking the value first in a , its reciprocal bijection is continuous, transforms a sum into a product and takes the value a in first . Algebraic properties [ modifier | Modifier and code ] For all strictly positive real a And b and for all real x And and :ax=exp\u2061(xln\u2061(a))=exln\u2061(a),ax=bxlogb\u2061(a),axy=(ax)y.{displaystyle a^{x}=exp(xln(a))={rm {e}}^{xln(a)},quad a^{x}=b^{xlog _{b}(a)},quad a^{xy}=left(a^{x}right)^{y}.} Applications exp a : x \u21a6 a x are morphisms of groups (ab\u00e9liens) of ( R , +) in ( R + *, \u00d7):ax+y=axay,a0=1,1ax=a\u2212x.{displaystyle a^{x+y}=a^{x}a^{y},quad a^{0}=1,quad {frac {1}{a^{x}}}=a^{-x}.} These morphisms constitute an isomorphic group to ( R + *, \u00d7) ( via  a \u21a6 exp a ) – So also to ( R , +):axbx=(ab)x,1x=1,1ax=(1a)x,a1=a.{displaystyle a^{x}b^{x}=(ab)^{x},quad 1^{x}=1,quad {frac {1}{a^{x}}}=left({frac {1}{a}}right)^{x},quad a^{1}=a.} Function study [ modifier | Modifier and code ] The basic exponential function a is indefinitely derivable on R and its derivative has the expression exp a\u2032 \u2061 ( x ) = ln \u2061 ( a ) exp a\u2061 ( x ) . {displaystyle exp _{a}'(x)=ln(a)exp _{a}(x).} Since the exponential function is always positive, the sign of its derivative depends only on the sign of ln( a ) . The function is therefore strictly increasing when the base a is strictly larger than first ; it is strictly decreasing when the base is less than first and constant if we took as a base a = 1 . The limits of the basic exponential function a depend on the position of a compared to first : and a > 1 SO lim x\u2192+\u221ea x= + \u221e etlim x\u2192\u2212\u221ea x= 0  ; {Displaystyle Lim _ {xto +Infty} a^{x} = +Infty quad {rm {et}} quad lim _ {xto -infty} a^{x} = 0 ~;} and a a x= 0 etlim x\u2192\u2212\u221ea x= + \u221e . {Displaystyle Lim _ {xto +Infty} a^{x} = 0quad {rm {et}} lim _ {xto -infty} a^{x} = +Infty.} The exponential function has a predictable behavior compared to the power function: in the event of indeterminacy in +\u221e , it is the exponential that prevails: For all real a > 1 And b , lim x\u2192+\u221eaxxb= + \u221e . {displaystyle lim _{xto +infty }{frac {a^{x}}{x^{b}}}=+infty .} It is both convex (therefore convex) and logarithmically concave (in) . 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