Why are $ln x$ and $e^x$ considered to be each others' inverses?











up vote
0
down vote

favorite












From my understanding the definition of a function's inverse is as follows.
Take a function $f$ which has the inverse $f^{-1}$. This would mean that $f(f^{-1}(x)) = x$ and that $f^{-1}(f(x)) = x$ for every real value for $x$. Right? And this is true for $ln(e^x) = x$ but for $e^{(ln x)} = x$ it only holds true for $x > 0$. Why are they still considered each others' inverse?






real-analysis inverse-function







share|cite|improve this question




















  • 1




    It's true because the range of $xmapsto e^x$ for $xin mathbb{R}$ is $(0, infty)$, so of course it only makes sense for the inverse to be defined on $(0, infty)$. Strictly speaking, your definition of inverse isn't quite right; the inverse of a function $f : Xto Y$ is defined $f^{-1} : Yto X$.
    – Michael Lee
    Nov 21 at 17:46












  • As @Micheal Lee explained, if f:A→B then it's inverse is f−1:B→A, and these sets are not necessarily the all the real numbers. Particularly, the exponential function is not surjective over the reals (hence not inversible) but it is surjective and injective over the positive reals.
    – Mefitico
    Nov 21 at 18:51















up vote
0
down vote

favorite












From my understanding the definition of a function's inverse is as follows.
Take a function $f$ which has the inverse $f^{-1}$. This would mean that $f(f^{-1}(x)) = x$ and that $f^{-1}(f(x)) = x$ for every real value for $x$. Right? And this is true for $ln(e^x) = x$ but for $e^{(ln x)} = x$ it only holds true for $x > 0$. Why are they still considered each others' inverse?






real-analysis inverse-function







share|cite|improve this question




















  • 1




    It's true because the range of $xmapsto e^x$ for $xin mathbb{R}$ is $(0, infty)$, so of course it only makes sense for the inverse to be defined on $(0, infty)$. Strictly speaking, your definition of inverse isn't quite right; the inverse of a function $f : Xto Y$ is defined $f^{-1} : Yto X$.
    – Michael Lee
    Nov 21 at 17:46












  • As @Micheal Lee explained, if f:A→B then it's inverse is f−1:B→A, and these sets are not necessarily the all the real numbers. Particularly, the exponential function is not surjective over the reals (hence not inversible) but it is surjective and injective over the positive reals.
    – Mefitico
    Nov 21 at 18:51













up vote
0
down vote

favorite









up vote
0
down vote

favorite











From my understanding the definition of a function's inverse is as follows.
Take a function $f$ which has the inverse $f^{-1}$. This would mean that $f(f^{-1}(x)) = x$ and that $f^{-1}(f(x)) = x$ for every real value for $x$. Right? And this is true for $ln(e^x) = x$ but for $e^{(ln x)} = x$ it only holds true for $x > 0$. Why are they still considered each others' inverse?






real-analysis inverse-function







share|cite|improve this question















From my understanding the definition of a function's inverse is as follows.
Take a function $f$ which has the inverse $f^{-1}$. This would mean that $f(f^{-1}(x)) = x$ and that $f^{-1}(f(x)) = x$ for every real value for $x$. Right? And this is true for $ln(e^x) = x$ but for $e^{(ln x)} = x$ it only holds true for $x > 0$. Why are they still considered each others' inverse?






real-analysis inverse-function




real-analysis inverse-function






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Nov 21 at 17:48









Rócherz

2,7162721




2,7162721










asked Nov 21 at 17:41









CrazyMineCoder

32




32








  • 1




    It's true because the range of $xmapsto e^x$ for $xin mathbb{R}$ is $(0, infty)$, so of course it only makes sense for the inverse to be defined on $(0, infty)$. Strictly speaking, your definition of inverse isn't quite right; the inverse of a function $f : Xto Y$ is defined $f^{-1} : Yto X$.
    – Michael Lee
    Nov 21 at 17:46












  • As @Micheal Lee explained, if f:A→B then it's inverse is f−1:B→A, and these sets are not necessarily the all the real numbers. Particularly, the exponential function is not surjective over the reals (hence not inversible) but it is surjective and injective over the positive reals.
    – Mefitico
    Nov 21 at 18:51














  • 1




    It's true because the range of $xmapsto e^x$ for $xin mathbb{R}$ is $(0, infty)$, so of course it only makes sense for the inverse to be defined on $(0, infty)$. Strictly speaking, your definition of inverse isn't quite right; the inverse of a function $f : Xto Y$ is defined $f^{-1} : Yto X$.
    – Michael Lee
    Nov 21 at 17:46












  • As @Micheal Lee explained, if f:A→B then it's inverse is f−1:B→A, and these sets are not necessarily the all the real numbers. Particularly, the exponential function is not surjective over the reals (hence not inversible) but it is surjective and injective over the positive reals.
    – Mefitico
    Nov 21 at 18:51








1




1




It's true because the range of $xmapsto e^x$ for $xin mathbb{R}$ is $(0, infty)$, so of course it only makes sense for the inverse to be defined on $(0, infty)$. Strictly speaking, your definition of inverse isn't quite right; the inverse of a function $f : Xto Y$ is defined $f^{-1} : Yto X$.
– Michael Lee
Nov 21 at 17:46






It's true because the range of $xmapsto e^x$ for $xin mathbb{R}$ is $(0, infty)$, so of course it only makes sense for the inverse to be defined on $(0, infty)$. Strictly speaking, your definition of inverse isn't quite right; the inverse of a function $f : Xto Y$ is defined $f^{-1} : Yto X$.
– Michael Lee
Nov 21 at 17:46














As @Micheal Lee explained, if f:A→B then it's inverse is f−1:B→A, and these sets are not necessarily the all the real numbers. Particularly, the exponential function is not surjective over the reals (hence not inversible) but it is surjective and injective over the positive reals.
– Mefitico
Nov 21 at 18:51




As @Micheal Lee explained, if f:A→B then it's inverse is f−1:B→A, and these sets are not necessarily the all the real numbers. Particularly, the exponential function is not surjective over the reals (hence not inversible) but it is surjective and injective over the positive reals.
– Mefitico
Nov 21 at 18:51










2 Answers
2






active

oldest

votes

















up vote
1
down vote



accepted










By definition, inverse functions have the other one’s domain and range.



The function $f(x) = e^x$ has a domain $x in mathbb{R}$ (all real numbers) and range of $y > 0$ (all positive numbers).



Therefore, $f^{-1}(x) = ln x$ has a domain $x > 0$ and range $y in mathbb{R}$.



$e^{ln x}$ being defined for $x > 0$ has to do with the domain of $ln x$.



$(fcirc f^{-1})(x) = x$ given that $x$ lies within the domain of $f^{-1}(x)$ and that $f^{-1}(x)$ lies within the domain of $f(x)$, which is the case here if $x>0$.






share|cite|improve this answer






























    up vote
    1
    down vote













    If you have two sets $A,B$ (they may be the same set, or they may not), and two functions $f:Ato B$ and $g:Bto A$, then $f$ and $g$ are said to be eachother's inverses if $g(f(a))=a$ for all $ain A$ and $f(g(b))=b$ for all $bin B$.



    In this case, your two sets are the set of real numbers $Bbb R$ and the set of positive real numbers $Bbb R^+$.






    share|cite|improve this answer





















      Your Answer





      StackExchange.ifUsing("editor", function () {
      return StackExchange.using("mathjaxEditing", function () {
      StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
      StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
      });
      });
      }, "mathjax-editing");

      StackExchange.ready(function() {
      var channelOptions = {
      tags: "".split(" "),
      id: "69"
      };
      initTagRenderer("".split(" "), "".split(" "), channelOptions);

      StackExchange.using("externalEditor", function() {
      // Have to fire editor after snippets, if snippets enabled
      if (StackExchange.settings.snippets.snippetsEnabled) {
      StackExchange.using("snippets", function() {
      createEditor();
      });
      }
      else {
      createEditor();
      }
      });

      function createEditor() {
      StackExchange.prepareEditor({
      heartbeatType: 'answer',
      convertImagesToLinks: true,
      noModals: true,
      showLowRepImageUploadWarning: true,
      reputationToPostImages: 10,
      bindNavPrevention: true,
      postfix: "",
      imageUploader: {
      brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
      contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
      allowUrls: true
      },
      noCode: true, onDemand: true,
      discardSelector: ".discard-answer"
      ,immediatelyShowMarkdownHelp:true
      });


      }
      });














      draft saved

      draft discarded


















      StackExchange.ready(
      function () {
      StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3008083%2fwhy-are-ln-x-and-ex-considered-to-be-each-others-inverses%23new-answer', 'question_page');
      }
      );

      Post as a guest















      Required, but never shown

























      2 Answers
      2






      active

      oldest

      votes








      2 Answers
      2






      active

      oldest

      votes









      active

      oldest

      votes






      active

      oldest

      votes








      up vote
      1
      down vote



      accepted










      By definition, inverse functions have the other one’s domain and range.



      The function $f(x) = e^x$ has a domain $x in mathbb{R}$ (all real numbers) and range of $y > 0$ (all positive numbers).



      Therefore, $f^{-1}(x) = ln x$ has a domain $x > 0$ and range $y in mathbb{R}$.



      $e^{ln x}$ being defined for $x > 0$ has to do with the domain of $ln x$.



      $(fcirc f^{-1})(x) = x$ given that $x$ lies within the domain of $f^{-1}(x)$ and that $f^{-1}(x)$ lies within the domain of $f(x)$, which is the case here if $x>0$.






      share|cite|improve this answer



























        up vote
        1
        down vote



        accepted










        By definition, inverse functions have the other one’s domain and range.



        The function $f(x) = e^x$ has a domain $x in mathbb{R}$ (all real numbers) and range of $y > 0$ (all positive numbers).



        Therefore, $f^{-1}(x) = ln x$ has a domain $x > 0$ and range $y in mathbb{R}$.



        $e^{ln x}$ being defined for $x > 0$ has to do with the domain of $ln x$.



        $(fcirc f^{-1})(x) = x$ given that $x$ lies within the domain of $f^{-1}(x)$ and that $f^{-1}(x)$ lies within the domain of $f(x)$, which is the case here if $x>0$.






        share|cite|improve this answer

























          up vote
          1
          down vote



          accepted







          up vote
          1
          down vote



          accepted






          By definition, inverse functions have the other one’s domain and range.



          The function $f(x) = e^x$ has a domain $x in mathbb{R}$ (all real numbers) and range of $y > 0$ (all positive numbers).



          Therefore, $f^{-1}(x) = ln x$ has a domain $x > 0$ and range $y in mathbb{R}$.



          $e^{ln x}$ being defined for $x > 0$ has to do with the domain of $ln x$.



          $(fcirc f^{-1})(x) = x$ given that $x$ lies within the domain of $f^{-1}(x)$ and that $f^{-1}(x)$ lies within the domain of $f(x)$, which is the case here if $x>0$.






          share|cite|improve this answer














          By definition, inverse functions have the other one’s domain and range.



          The function $f(x) = e^x$ has a domain $x in mathbb{R}$ (all real numbers) and range of $y > 0$ (all positive numbers).



          Therefore, $f^{-1}(x) = ln x$ has a domain $x > 0$ and range $y in mathbb{R}$.



          $e^{ln x}$ being defined for $x > 0$ has to do with the domain of $ln x$.



          $(fcirc f^{-1})(x) = x$ given that $x$ lies within the domain of $f^{-1}(x)$ and that $f^{-1}(x)$ lies within the domain of $f(x)$, which is the case here if $x>0$.







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited Nov 21 at 18:10

























          answered Nov 21 at 17:47









          KM101

          3,609417




          3,609417






















              up vote
              1
              down vote













              If you have two sets $A,B$ (they may be the same set, or they may not), and two functions $f:Ato B$ and $g:Bto A$, then $f$ and $g$ are said to be eachother's inverses if $g(f(a))=a$ for all $ain A$ and $f(g(b))=b$ for all $bin B$.



              In this case, your two sets are the set of real numbers $Bbb R$ and the set of positive real numbers $Bbb R^+$.






              share|cite|improve this answer

























                up vote
                1
                down vote













                If you have two sets $A,B$ (they may be the same set, or they may not), and two functions $f:Ato B$ and $g:Bto A$, then $f$ and $g$ are said to be eachother's inverses if $g(f(a))=a$ for all $ain A$ and $f(g(b))=b$ for all $bin B$.



                In this case, your two sets are the set of real numbers $Bbb R$ and the set of positive real numbers $Bbb R^+$.






                share|cite|improve this answer























                  up vote
                  1
                  down vote










                  up vote
                  1
                  down vote









                  If you have two sets $A,B$ (they may be the same set, or they may not), and two functions $f:Ato B$ and $g:Bto A$, then $f$ and $g$ are said to be eachother's inverses if $g(f(a))=a$ for all $ain A$ and $f(g(b))=b$ for all $bin B$.



                  In this case, your two sets are the set of real numbers $Bbb R$ and the set of positive real numbers $Bbb R^+$.






                  share|cite|improve this answer












                  If you have two sets $A,B$ (they may be the same set, or they may not), and two functions $f:Ato B$ and $g:Bto A$, then $f$ and $g$ are said to be eachother's inverses if $g(f(a))=a$ for all $ain A$ and $f(g(b))=b$ for all $bin B$.



                  In this case, your two sets are the set of real numbers $Bbb R$ and the set of positive real numbers $Bbb R^+$.







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered Nov 21 at 17:47









                  Arthur

                  110k7104186




                  110k7104186






























                      draft saved

                      draft discarded




















































                      Thanks for contributing an answer to Mathematics Stack Exchange!


                      • Please be sure to answer the question. Provide details and share your research!

                      But avoid …



                      • Asking for help, clarification, or responding to other answers.

                      • Making statements based on opinion; back them up with references or personal experience.


                      Use MathJax to format equations. MathJax reference.


                      To learn more, see our tips on writing great answers.





                      Some of your past answers have not been well-received, and you're in danger of being blocked from answering.


                      Please pay close attention to the following guidance:


                      • Please be sure to answer the question. Provide details and share your research!

                      But avoid …



                      • Asking for help, clarification, or responding to other answers.

                      • Making statements based on opinion; back them up with references or personal experience.


                      To learn more, see our tips on writing great answers.




                      draft saved


                      draft discarded














                      StackExchange.ready(
                      function () {
                      StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3008083%2fwhy-are-ln-x-and-ex-considered-to-be-each-others-inverses%23new-answer', 'question_page');
                      }
                      );

                      Post as a guest















                      Required, but never shown





















































                      Required, but never shown














                      Required, but never shown












                      Required, but never shown







                      Required, but never shown

































                      Required, but never shown














                      Required, but never shown












                      Required, but never shown







                      Required, but never shown







                      -

                      Popular posts from this blog

                      Ellipse (mathématiques)

                      Image
                      Pour les articles homonymes, voir Ellipse. L'ellipse est le lieu des points dont la somme des distances à deux points fixes, dits foyers, est constante. Section du cône ou projection du cercle. En géométrie, une ellipse est une courbe plane fermée obtenue par l’intersection d’un cône de révolution avec un plan, à condition que celui-ci coupe l'axe de rotation du cône ou du cylindre : c'est une conique d'excentricité strictement comprise entre 0 et 1. On peut également la définir comme le lieu des points dont la somme des distances à deux points fixes, dits foyers, est constante (sa construction par la méthode du jardinier est très simple). Dans la vie courante, l’ellipse est la forme qu'on perçoit en regardant un cercle en perspective, ou la figure formée par l’ombre d'un disque sur une surface plane. On retrouve aussi, en première approximation [ 1 ] , des ellipses dans les trajectoires des corps célestes (planètes, comètes ou satellites arti...
                      Read more

                      Quarter-circle Tiles

                      Image
                      up vote 0 down vote favorite I have $99$ identical square tiles, each with a quarter-circle drawn on it like this: [asy] size(1.5cm); draw(Arc((2,0),1,90,180),red+1); draw((0,0)--(2,0)--(2,2)--(0,2)--(0,0)); [/asy] When I arrange the tiles in a $9times 11$ rectangular grid, each with a random orientation, what is the expected value of the number of full circles I form? I think this problem has to do with finding the chance any given 2x2 square has a circle, but I can't find it. expected-value share | cite | improve this question asked Nov 20 at 15:03 6minecraftninja 1 2 ...
                      Read more

                      Mont Emei

                      Image
                      Mont Emei.mw-parser-output .entete.map{background-image:url("//upload.wikimedia.org/wikipedia/commons/7/7a/Picto_infobox_map.png")} Vue du sommet Wanfo du mont Emei Géographie Altitude 3 099  m , Wanfo Coordonnées 29° 31′ 11″ nord, 103° 19′ 57″ est Administration Pays   Chine Province Sichuan Ville-préfecture Leshan Géolocalisation sur la carte : Sichuan Mont Emei Géolocalisation sur la carte : Chine Mont Emei modifier   Paysage panoramique du mont Emei, incluant le paysage panoramique du grand Bouddha de Leshan *   Patrimoine mondial de l'UNESCO Passerelle en bois sur le versant ouest Pays   Chine Subdivision Sichuan Type Mixte Critères (iv) (vi) (x) Superficie 15 400 ha Numéro d’identification 779 Zone géographique Asie et Pacifique  ** Année d’inscription 1996 (20 e session) * Descriptif officiel UNESCO ** Classification géog...
                      Read more