Let $P$ be the group of strictly positive real numbers under multiplication. Prove that $P$ isomorphic to...












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Let $P$ be the group of strictly positive real numbers under multiplication. Prove that $P$ isomorphic to $(mathbb{R}, +)$






group-theory group-isomorphism







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closed as off-topic by M. Winter, verret, Gibbs, Lee David Chung Lin, Lord Shark the Unknown Jan 29 at 5:19


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – M. Winter, verret, Gibbs, Lee David Chung Lin, Lord Shark the Unknown

If this question can be reworded to fit the rules in the help center, please edit the question.





















    2












    $begingroup$


    Let $P$ be the group of strictly positive real numbers under multiplication. Prove that $P$ isomorphic to $(mathbb{R}, +)$






    group-theory group-isomorphism







    share|cite|improve this question











    $endgroup$



    closed as off-topic by M. Winter, verret, Gibbs, Lee David Chung Lin, Lord Shark the Unknown Jan 29 at 5:19


    This question appears to be off-topic. The users who voted to close gave this specific reason:


    • "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – M. Winter, verret, Gibbs, Lee David Chung Lin, Lord Shark the Unknown

    If this question can be reworded to fit the rules in the help center, please edit the question.



















      2












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      2


      0



      $begingroup$


      Let $P$ be the group of strictly positive real numbers under multiplication. Prove that $P$ isomorphic to $(mathbb{R}, +)$






      group-theory group-isomorphism







      share|cite|improve this question











      $endgroup$




      Let $P$ be the group of strictly positive real numbers under multiplication. Prove that $P$ isomorphic to $(mathbb{R}, +)$






      group-theory group-isomorphism




      group-theory group-isomorphism






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Jan 28 at 16:17









      idriskameni

      732321




      732321










      asked Jan 28 at 15:49









      Grace3001Grace3001

      183




      183




      closed as off-topic by M. Winter, verret, Gibbs, Lee David Chung Lin, Lord Shark the Unknown Jan 29 at 5:19


      This question appears to be off-topic. The users who voted to close gave this specific reason:


      • "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – M. Winter, verret, Gibbs, Lee David Chung Lin, Lord Shark the Unknown

      If this question can be reworded to fit the rules in the help center, please edit the question.







      closed as off-topic by M. Winter, verret, Gibbs, Lee David Chung Lin, Lord Shark the Unknown Jan 29 at 5:19


      This question appears to be off-topic. The users who voted to close gave this specific reason:


      • "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – M. Winter, verret, Gibbs, Lee David Chung Lin, Lord Shark the Unknown

      If this question can be reworded to fit the rules in the help center, please edit the question.






















          2 Answers
          2






          active

          oldest

          votes


















          2












          $begingroup$

          Use $$phi: ,x mapsto e^x$$
          which is a group homomorphism thanks to the exponential properties.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            Okay thank you I think I know how to approach it now !
            $endgroup$
            – Grace3001
            Jan 28 at 15:53










          • $begingroup$
            You're welcome!
            $endgroup$
            – Harnak
            Jan 28 at 16:04



















          2












          $begingroup$

          Hint: Consider the exponential function $e:{Bbb R}rightarrow {Bbb R}_{>0}:xmapsto e^x$ and its inverse $log_e :{Bbb R}_{>0}rightarrow {Bbb R}:xmapsto log_e x$.



          We have $log_e(e^x) = x$, $e^{log_e x} = x$, $log_e(xcdot y) = log_e(x) + log_e(y)$ and $e^{x+y} = e^xcdot e^y$.






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            Are their any other isomorphisms between $(mathbb{R}, +)$ and $(mathbb{R}^+, cdot)$?
            $endgroup$
            – feynhat
            Jan 28 at 16:00




















          2 Answers
          2






          active

          oldest

          votes








          2 Answers
          2






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes









          2












          $begingroup$

          Use $$phi: ,x mapsto e^x$$
          which is a group homomorphism thanks to the exponential properties.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            Okay thank you I think I know how to approach it now !
            $endgroup$
            – Grace3001
            Jan 28 at 15:53










          • $begingroup$
            You're welcome!
            $endgroup$
            – Harnak
            Jan 28 at 16:04
















          2












          $begingroup$

          Use $$phi: ,x mapsto e^x$$
          which is a group homomorphism thanks to the exponential properties.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            Okay thank you I think I know how to approach it now !
            $endgroup$
            – Grace3001
            Jan 28 at 15:53










          • $begingroup$
            You're welcome!
            $endgroup$
            – Harnak
            Jan 28 at 16:04














          2












          2








          2





          $begingroup$

          Use $$phi: ,x mapsto e^x$$
          which is a group homomorphism thanks to the exponential properties.






          share|cite|improve this answer









          $endgroup$



          Use $$phi: ,x mapsto e^x$$
          which is a group homomorphism thanks to the exponential properties.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Jan 28 at 15:53









          HarnakHarnak

          1,309512




          1,309512












          • $begingroup$
            Okay thank you I think I know how to approach it now !
            $endgroup$
            – Grace3001
            Jan 28 at 15:53










          • $begingroup$
            You're welcome!
            $endgroup$
            – Harnak
            Jan 28 at 16:04


















          • $begingroup$
            Okay thank you I think I know how to approach it now !
            $endgroup$
            – Grace3001
            Jan 28 at 15:53










          • $begingroup$
            You're welcome!
            $endgroup$
            – Harnak
            Jan 28 at 16:04
















          $begingroup$
          Okay thank you I think I know how to approach it now !
          $endgroup$
          – Grace3001
          Jan 28 at 15:53




          $begingroup$
          Okay thank you I think I know how to approach it now !
          $endgroup$
          – Grace3001
          Jan 28 at 15:53












          $begingroup$
          You're welcome!
          $endgroup$
          – Harnak
          Jan 28 at 16:04




          $begingroup$
          You're welcome!
          $endgroup$
          – Harnak
          Jan 28 at 16:04











          2












          $begingroup$

          Hint: Consider the exponential function $e:{Bbb R}rightarrow {Bbb R}_{>0}:xmapsto e^x$ and its inverse $log_e :{Bbb R}_{>0}rightarrow {Bbb R}:xmapsto log_e x$.



          We have $log_e(e^x) = x$, $e^{log_e x} = x$, $log_e(xcdot y) = log_e(x) + log_e(y)$ and $e^{x+y} = e^xcdot e^y$.






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            Are their any other isomorphisms between $(mathbb{R}, +)$ and $(mathbb{R}^+, cdot)$?
            $endgroup$
            – feynhat
            Jan 28 at 16:00


















          2












          $begingroup$

          Hint: Consider the exponential function $e:{Bbb R}rightarrow {Bbb R}_{>0}:xmapsto e^x$ and its inverse $log_e :{Bbb R}_{>0}rightarrow {Bbb R}:xmapsto log_e x$.



          We have $log_e(e^x) = x$, $e^{log_e x} = x$, $log_e(xcdot y) = log_e(x) + log_e(y)$ and $e^{x+y} = e^xcdot e^y$.






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            Are their any other isomorphisms between $(mathbb{R}, +)$ and $(mathbb{R}^+, cdot)$?
            $endgroup$
            – feynhat
            Jan 28 at 16:00
















          2












          2








          2





          $begingroup$

          Hint: Consider the exponential function $e:{Bbb R}rightarrow {Bbb R}_{>0}:xmapsto e^x$ and its inverse $log_e :{Bbb R}_{>0}rightarrow {Bbb R}:xmapsto log_e x$.



          We have $log_e(e^x) = x$, $e^{log_e x} = x$, $log_e(xcdot y) = log_e(x) + log_e(y)$ and $e^{x+y} = e^xcdot e^y$.






          share|cite|improve this answer











          $endgroup$



          Hint: Consider the exponential function $e:{Bbb R}rightarrow {Bbb R}_{>0}:xmapsto e^x$ and its inverse $log_e :{Bbb R}_{>0}rightarrow {Bbb R}:xmapsto log_e x$.



          We have $log_e(e^x) = x$, $e^{log_e x} = x$, $log_e(xcdot y) = log_e(x) + log_e(y)$ and $e^{x+y} = e^xcdot e^y$.







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited Jan 28 at 18:39

























          answered Jan 28 at 15:52









          WuestenfuxWuestenfux

          5,3231513




          5,3231513












          • $begingroup$
            Are their any other isomorphisms between $(mathbb{R}, +)$ and $(mathbb{R}^+, cdot)$?
            $endgroup$
            – feynhat
            Jan 28 at 16:00




















          • $begingroup$
            Are their any other isomorphisms between $(mathbb{R}, +)$ and $(mathbb{R}^+, cdot)$?
            $endgroup$
            – feynhat
            Jan 28 at 16:00


















          $begingroup$
          Are their any other isomorphisms between $(mathbb{R}, +)$ and $(mathbb{R}^+, cdot)$?
          $endgroup$
          – feynhat
          Jan 28 at 16:00






          $begingroup$
          Are their any other isomorphisms between $(mathbb{R}, +)$ and $(mathbb{R}^+, cdot)$?
          $endgroup$
          – feynhat
          Jan 28 at 16:00





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