Problem with system of logarythmic equation.












1












$begingroup$


So I have this problem with a system of logarytmic equations. Specifically how to get rid of the (ln(x))^2 in order to solve this one.
I know you can get This as a solution for the first equation.
My problem is how to use this in the second one.
I can also provide the solution for the problem, however I don't know which steps to do how to get there.Solution fo the problem



Thanks in advance



ln(x) + ln(y^2)=4 ;
(ln(x))^2 - 3ln(xy)=-5






logarithms systems-of-equations nonlinear-system







share|cite|improve this question









$endgroup$

















    1












    $begingroup$


    So I have this problem with a system of logarytmic equations. Specifically how to get rid of the (ln(x))^2 in order to solve this one.
    I know you can get This as a solution for the first equation.
    My problem is how to use this in the second one.
    I can also provide the solution for the problem, however I don't know which steps to do how to get there.Solution fo the problem



    Thanks in advance



    ln(x) + ln(y^2)=4 ;
    (ln(x))^2 - 3ln(xy)=-5






    logarithms systems-of-equations nonlinear-system







    share|cite|improve this question









    $endgroup$















      1












      1








      1





      $begingroup$


      So I have this problem with a system of logarytmic equations. Specifically how to get rid of the (ln(x))^2 in order to solve this one.
      I know you can get This as a solution for the first equation.
      My problem is how to use this in the second one.
      I can also provide the solution for the problem, however I don't know which steps to do how to get there.Solution fo the problem



      Thanks in advance



      ln(x) + ln(y^2)=4 ;
      (ln(x))^2 - 3ln(xy)=-5






      logarithms systems-of-equations nonlinear-system







      share|cite|improve this question









      $endgroup$




      So I have this problem with a system of logarytmic equations. Specifically how to get rid of the (ln(x))^2 in order to solve this one.
      I know you can get This as a solution for the first equation.
      My problem is how to use this in the second one.
      I can also provide the solution for the problem, however I don't know which steps to do how to get there.Solution fo the problem



      Thanks in advance



      ln(x) + ln(y^2)=4 ;
      (ln(x))^2 - 3ln(xy)=-5






      logarithms systems-of-equations nonlinear-system




      logarithms systems-of-equations nonlinear-system






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Dec 30 '18 at 9:54









      user630424user630424

      82




      82






















          2 Answers
          2






          active

          oldest

          votes


















          0












          $begingroup$

          $x>0$ and from here $y>0$.



          Let $ln{y}=t$.



          Thus, $ln{x}=4-2t$ and we obtain:
          $$(4-2t)^2-3(2-2t)-3t=-5$$ or
          $$4t^2-13t+9=0.$$
          Can you end it now?



          I got the following answer.
          $$left{(e^2,e),left(frac{1}{sqrt{e}},e^{frac{9}{4}}right)right}$$






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            Yes! Thank you very much Sir! I can get that same solution now.
            $endgroup$
            – user630424
            Dec 30 '18 at 10:10










          • $begingroup$
            @user630424 You are welcome!
            $endgroup$
            – Michael Rozenberg
            Dec 30 '18 at 10:11



















          0












          $begingroup$

          Assume $x$ and $y$ to be positive. Then
          $$log()+log(y^2)=4 implies log(x)+2log(y)=4 tag1$$
          $$log^2(x)-3log(xy)=5 implies log^2(x)-3 log(x)-3log(y)=5 tag 2$$



          From $(1)$, express $log(y)$ as a function of $log(x)$. Replace in $(2)$ and you have a quadratic equation in $log(x)$.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            I see, when we then substitute ln(x) = t we get the same equation as Michael's. Thank you vey much for your help!
            $endgroup$
            – user630424
            Dec 30 '18 at 10:12










          • $begingroup$
            @user630424. You are welcome !
            $endgroup$
            – Claude Leibovici
            Dec 30 '18 at 10:28











          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',
          autoActivateHeartbeat: false,
          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%2f3056664%2fproblem-with-system-of-logarythmic-equation%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









          0












          $begingroup$

          $x>0$ and from here $y>0$.



          Let $ln{y}=t$.



          Thus, $ln{x}=4-2t$ and we obtain:
          $$(4-2t)^2-3(2-2t)-3t=-5$$ or
          $$4t^2-13t+9=0.$$
          Can you end it now?



          I got the following answer.
          $$left{(e^2,e),left(frac{1}{sqrt{e}},e^{frac{9}{4}}right)right}$$






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            Yes! Thank you very much Sir! I can get that same solution now.
            $endgroup$
            – user630424
            Dec 30 '18 at 10:10










          • $begingroup$
            @user630424 You are welcome!
            $endgroup$
            – Michael Rozenberg
            Dec 30 '18 at 10:11
















          0












          $begingroup$

          $x>0$ and from here $y>0$.



          Let $ln{y}=t$.



          Thus, $ln{x}=4-2t$ and we obtain:
          $$(4-2t)^2-3(2-2t)-3t=-5$$ or
          $$4t^2-13t+9=0.$$
          Can you end it now?



          I got the following answer.
          $$left{(e^2,e),left(frac{1}{sqrt{e}},e^{frac{9}{4}}right)right}$$






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            Yes! Thank you very much Sir! I can get that same solution now.
            $endgroup$
            – user630424
            Dec 30 '18 at 10:10










          • $begingroup$
            @user630424 You are welcome!
            $endgroup$
            – Michael Rozenberg
            Dec 30 '18 at 10:11














          0












          0








          0





          $begingroup$

          $x>0$ and from here $y>0$.



          Let $ln{y}=t$.



          Thus, $ln{x}=4-2t$ and we obtain:
          $$(4-2t)^2-3(2-2t)-3t=-5$$ or
          $$4t^2-13t+9=0.$$
          Can you end it now?



          I got the following answer.
          $$left{(e^2,e),left(frac{1}{sqrt{e}},e^{frac{9}{4}}right)right}$$






          share|cite|improve this answer









          $endgroup$



          $x>0$ and from here $y>0$.



          Let $ln{y}=t$.



          Thus, $ln{x}=4-2t$ and we obtain:
          $$(4-2t)^2-3(2-2t)-3t=-5$$ or
          $$4t^2-13t+9=0.$$
          Can you end it now?



          I got the following answer.
          $$left{(e^2,e),left(frac{1}{sqrt{e}},e^{frac{9}{4}}right)right}$$







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Dec 30 '18 at 9:59









          Michael RozenbergMichael Rozenberg

          105k1892198




          105k1892198












          • $begingroup$
            Yes! Thank you very much Sir! I can get that same solution now.
            $endgroup$
            – user630424
            Dec 30 '18 at 10:10










          • $begingroup$
            @user630424 You are welcome!
            $endgroup$
            – Michael Rozenberg
            Dec 30 '18 at 10:11


















          • $begingroup$
            Yes! Thank you very much Sir! I can get that same solution now.
            $endgroup$
            – user630424
            Dec 30 '18 at 10:10










          • $begingroup$
            @user630424 You are welcome!
            $endgroup$
            – Michael Rozenberg
            Dec 30 '18 at 10:11
















          $begingroup$
          Yes! Thank you very much Sir! I can get that same solution now.
          $endgroup$
          – user630424
          Dec 30 '18 at 10:10




          $begingroup$
          Yes! Thank you very much Sir! I can get that same solution now.
          $endgroup$
          – user630424
          Dec 30 '18 at 10:10












          $begingroup$
          @user630424 You are welcome!
          $endgroup$
          – Michael Rozenberg
          Dec 30 '18 at 10:11




          $begingroup$
          @user630424 You are welcome!
          $endgroup$
          – Michael Rozenberg
          Dec 30 '18 at 10:11











          0












          $begingroup$

          Assume $x$ and $y$ to be positive. Then
          $$log()+log(y^2)=4 implies log(x)+2log(y)=4 tag1$$
          $$log^2(x)-3log(xy)=5 implies log^2(x)-3 log(x)-3log(y)=5 tag 2$$



          From $(1)$, express $log(y)$ as a function of $log(x)$. Replace in $(2)$ and you have a quadratic equation in $log(x)$.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            I see, when we then substitute ln(x) = t we get the same equation as Michael's. Thank you vey much for your help!
            $endgroup$
            – user630424
            Dec 30 '18 at 10:12










          • $begingroup$
            @user630424. You are welcome !
            $endgroup$
            – Claude Leibovici
            Dec 30 '18 at 10:28
















          0












          $begingroup$

          Assume $x$ and $y$ to be positive. Then
          $$log()+log(y^2)=4 implies log(x)+2log(y)=4 tag1$$
          $$log^2(x)-3log(xy)=5 implies log^2(x)-3 log(x)-3log(y)=5 tag 2$$



          From $(1)$, express $log(y)$ as a function of $log(x)$. Replace in $(2)$ and you have a quadratic equation in $log(x)$.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            I see, when we then substitute ln(x) = t we get the same equation as Michael's. Thank you vey much for your help!
            $endgroup$
            – user630424
            Dec 30 '18 at 10:12










          • $begingroup$
            @user630424. You are welcome !
            $endgroup$
            – Claude Leibovici
            Dec 30 '18 at 10:28














          0












          0








          0





          $begingroup$

          Assume $x$ and $y$ to be positive. Then
          $$log()+log(y^2)=4 implies log(x)+2log(y)=4 tag1$$
          $$log^2(x)-3log(xy)=5 implies log^2(x)-3 log(x)-3log(y)=5 tag 2$$



          From $(1)$, express $log(y)$ as a function of $log(x)$. Replace in $(2)$ and you have a quadratic equation in $log(x)$.






          share|cite|improve this answer









          $endgroup$



          Assume $x$ and $y$ to be positive. Then
          $$log()+log(y^2)=4 implies log(x)+2log(y)=4 tag1$$
          $$log^2(x)-3log(xy)=5 implies log^2(x)-3 log(x)-3log(y)=5 tag 2$$



          From $(1)$, express $log(y)$ as a function of $log(x)$. Replace in $(2)$ and you have a quadratic equation in $log(x)$.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Dec 30 '18 at 10:03









          Claude LeiboviciClaude Leibovici

          122k1157134




          122k1157134












          • $begingroup$
            I see, when we then substitute ln(x) = t we get the same equation as Michael's. Thank you vey much for your help!
            $endgroup$
            – user630424
            Dec 30 '18 at 10:12










          • $begingroup$
            @user630424. You are welcome !
            $endgroup$
            – Claude Leibovici
            Dec 30 '18 at 10:28


















          • $begingroup$
            I see, when we then substitute ln(x) = t we get the same equation as Michael's. Thank you vey much for your help!
            $endgroup$
            – user630424
            Dec 30 '18 at 10:12










          • $begingroup$
            @user630424. You are welcome !
            $endgroup$
            – Claude Leibovici
            Dec 30 '18 at 10:28
















          $begingroup$
          I see, when we then substitute ln(x) = t we get the same equation as Michael's. Thank you vey much for your help!
          $endgroup$
          – user630424
          Dec 30 '18 at 10:12




          $begingroup$
          I see, when we then substitute ln(x) = t we get the same equation as Michael's. Thank you vey much for your help!
          $endgroup$
          – user630424
          Dec 30 '18 at 10:12












          $begingroup$
          @user630424. You are welcome !
          $endgroup$
          – Claude Leibovici
          Dec 30 '18 at 10:28




          $begingroup$
          @user630424. You are welcome !
          $endgroup$
          – Claude Leibovici
          Dec 30 '18 at 10:28


















          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.




          draft saved


          draft discarded














          StackExchange.ready(
          function () {
          StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3056664%2fproblem-with-system-of-logarythmic-equation%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

          Bressuire

          Image
          .mw-parser-output .infobox{border:1px solid #aaa;background-color:#f9f9f9;color:black;margin:.5em 0 .5em 1em;padding:.2em;float:right;clear:right;width:22em;text-align:left;font-size:88%;line-height:1.6em}.mw-parser-output .infobox td,.mw-parser-output .infobox th{vertical-align:top;padding:0 .2em}.mw-parser-output .infobox caption{font-size:larger}.mw-parser-output .infobox.bordered{border-collapse:collapse}.mw-parser-output .infobox.bordered td,.mw-parser-output .infobox.bordered th{border:1px solid #aaa}.mw-parser-output .infobox.bordered .borderless td,.mw-parser-output .infobox.bordered .borderless th{border:0}.mw-parser-output .infobox-showbutton .mw-collapsible-text{color:inherit}.mw-parser-output .infobox.bordered .mergedtoprow td,.mw-parser-output .infobox.bordered .mergedtoprow th{border:0;border-top:1px solid #aaa;border-right:1px solid #aaa}.mw-parser-output .infobox.bordered .mergedrow td,.mw-parser-output .infobox.bordered .mergedrow th{border:0;border-right:1px solid ...
          Read more

          Cabo Verde

          Image
            Nota: Cabo-verdiano redireciona para este artigo. Para a língua falada em Cabo Verde, veja crioulo cabo-verdiano. Para outros significados, veja Cabo Verde (desambiguação). Coordenadas: 14° 55' N 23° 31' O Cabo Verde República de Cabo Verde Bandeira Brasão de Armas Lema: Unidade, Trabalho, Progresso Hino nacional: Cântico da Liberdade Gentílico: cabo-verdiano [ 1 ] Capital Praia 14° 55' N 23° 31' O Cidade mais populosa Praia Língua oficial Português Governo República unitária semipresidencialista [ 2 ]  - Presidente Jorge Carlos Fonseca  - Primeiro-ministro Ulisses Correia e Silva História    - Descoberta 1460   - Independência 00 • de Portugal 5 de Julho de 1975   - Multipartidarismo 13 de Janeiro de 1990  Área    - Total 4 033 km² (146.º)  - Água (%) 0,08 População    - Estimativa para 2017 560 899 [ 3 ] hab. (173.º) ...
          Read more

          Gyllenstierna

          Image
          .mw-parser-output .infobox{border:1px solid #aaa;background-color:#f9f9f9;color:black;margin:.5em 0 .5em 1em;padding:.2em;float:right;clear:right;width:22em;text-align:left;font-size:88%;line-height:1.6em}.mw-parser-output .infobox td,.mw-parser-output .infobox th{vertical-align:top;padding:0 .2em}.mw-parser-output .infobox caption{font-size:larger}.mw-parser-output .infobox.bordered{border-collapse:collapse}.mw-parser-output .infobox.bordered td,.mw-parser-output .infobox.bordered th{border:1px solid #aaa}.mw-parser-output .infobox.bordered .borderless td,.mw-parser-output .infobox.bordered .borderless th{border:0}.mw-parser-output .infobox-showbutton .mw-collapsible-text{color:inherit}.mw-parser-output .infobox.bordered .mergedtoprow td,.mw-parser-output .infobox.bordered .mergedtoprow th{border:0;border-top:1px solid #aaa;border-right:1px solid #aaa}.mw-parser-output .infobox.bordered .mergedrow td,.mw-parser-output .infobox.bordered .mergedrow th{border:0;border-right:1px solid ...
          Read more