How to solve $3^x=3-x$











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Is it possible to solve for $$3^x=3-x$$ without graphing it?



This question is in the section: Solving Exponential equations in my math textbook.



I have tried to log both sides and solve for it however that just leads me back to the original equation.






algebra-precalculus logarithms exponential-function







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  • Welcome to MSE. It will be more likely that you will get an answer if you show us that you made an effort. This should be added to the question rather than in the comments.
    – José Carlos Santos
    14 hours ago










  • Thank you sir. I like your feedback. Pardon me if I did not follow format, english is not my first language. I hope sir you understand.
    – warjwar8
    13 hours ago










  • Have you tried to solve $3^x + x = 3$?
    – Robert Soupe
    5 hours ago















up vote
1
down vote

favorite












Is it possible to solve for $$3^x=3-x$$ without graphing it?



This question is in the section: Solving Exponential equations in my math textbook.



I have tried to log both sides and solve for it however that just leads me back to the original equation.






algebra-precalculus logarithms exponential-function







share|cite|improve this question









New contributor




warjwar8 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.




















  • Welcome to MSE. It will be more likely that you will get an answer if you show us that you made an effort. This should be added to the question rather than in the comments.
    – José Carlos Santos
    14 hours ago










  • Thank you sir. I like your feedback. Pardon me if I did not follow format, english is not my first language. I hope sir you understand.
    – warjwar8
    13 hours ago










  • Have you tried to solve $3^x + x = 3$?
    – Robert Soupe
    5 hours ago













up vote
1
down vote

favorite









up vote
1
down vote

favorite











Is it possible to solve for $$3^x=3-x$$ without graphing it?



This question is in the section: Solving Exponential equations in my math textbook.



I have tried to log both sides and solve for it however that just leads me back to the original equation.






algebra-precalculus logarithms exponential-function







share|cite|improve this question









New contributor




warjwar8 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.











Is it possible to solve for $$3^x=3-x$$ without graphing it?



This question is in the section: Solving Exponential equations in my math textbook.



I have tried to log both sides and solve for it however that just leads me back to the original equation.






algebra-precalculus logarithms exponential-function




algebra-precalculus logarithms exponential-function






share|cite|improve this question









New contributor




warjwar8 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.











share|cite|improve this question









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warjwar8 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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share|cite|improve this question




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edited 14 hours ago





















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asked 14 hours ago









warjwar8

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warjwar8 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.






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Check out our Code of Conduct.












  • Welcome to MSE. It will be more likely that you will get an answer if you show us that you made an effort. This should be added to the question rather than in the comments.
    – José Carlos Santos
    14 hours ago










  • Thank you sir. I like your feedback. Pardon me if I did not follow format, english is not my first language. I hope sir you understand.
    – warjwar8
    13 hours ago










  • Have you tried to solve $3^x + x = 3$?
    – Robert Soupe
    5 hours ago


















  • Welcome to MSE. It will be more likely that you will get an answer if you show us that you made an effort. This should be added to the question rather than in the comments.
    – José Carlos Santos
    14 hours ago










  • Thank you sir. I like your feedback. Pardon me if I did not follow format, english is not my first language. I hope sir you understand.
    – warjwar8
    13 hours ago










  • Have you tried to solve $3^x + x = 3$?
    – Robert Soupe
    5 hours ago
















Welcome to MSE. It will be more likely that you will get an answer if you show us that you made an effort. This should be added to the question rather than in the comments.
– José Carlos Santos
14 hours ago




Welcome to MSE. It will be more likely that you will get an answer if you show us that you made an effort. This should be added to the question rather than in the comments.
– José Carlos Santos
14 hours ago












Thank you sir. I like your feedback. Pardon me if I did not follow format, english is not my first language. I hope sir you understand.
– warjwar8
13 hours ago




Thank you sir. I like your feedback. Pardon me if I did not follow format, english is not my first language. I hope sir you understand.
– warjwar8
13 hours ago












Have you tried to solve $3^x + x = 3$?
– Robert Soupe
5 hours ago




Have you tried to solve $3^x + x = 3$?
– Robert Soupe
5 hours ago










2 Answers
2






active

oldest

votes

















up vote
1
down vote













HINT



Let consider



$$f(x)=3^x-(3-x)$$



and note that




  • $f(0)=-2$

  • $f(1)=1$


therefore by IVT there is a solution in $(0,1)$.



Can you show that this is the unique solution?






share|cite|improve this answer





















  • If I am correct then: for $x>1$ the term $3^x +x$ is more positive than '3' everywhere in that domain. Also, for x<0 the term $-3+x$ is more negative to be neutralized by $3^x$ as $3^x$ will then become in postive fraction number.
    – jayant98
    13 hours ago






  • 1




    @jayant98 We can consider the sign of $f'(x)=3^x log 3 +1$.
    – gimusi
    13 hours ago








  • 1




    Yes Sir! Another way which is more simple and effective.
    – jayant98
    13 hours ago










  • @jayant98 Call me simply gimusi! Bye
    – gimusi
    13 hours ago


















up vote
1
down vote













Sure. The solution can be expressed in terms of the Lambert W function. In general, graphing is not a way of "solving" an equation because it cannot give you an exact solution. Graphing the equations above will help you approximate $x approx 0.742$ but cannot give you an exact value.






share|cite|improve this answer





















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    2 Answers
    2






    active

    oldest

    votes








    2 Answers
    2






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes








    up vote
    1
    down vote













    HINT



    Let consider



    $$f(x)=3^x-(3-x)$$



    and note that




    • $f(0)=-2$

    • $f(1)=1$


    therefore by IVT there is a solution in $(0,1)$.



    Can you show that this is the unique solution?






    share|cite|improve this answer





















    • If I am correct then: for $x>1$ the term $3^x +x$ is more positive than '3' everywhere in that domain. Also, for x<0 the term $-3+x$ is more negative to be neutralized by $3^x$ as $3^x$ will then become in postive fraction number.
      – jayant98
      13 hours ago






    • 1




      @jayant98 We can consider the sign of $f'(x)=3^x log 3 +1$.
      – gimusi
      13 hours ago








    • 1




      Yes Sir! Another way which is more simple and effective.
      – jayant98
      13 hours ago










    • @jayant98 Call me simply gimusi! Bye
      – gimusi
      13 hours ago















    up vote
    1
    down vote













    HINT



    Let consider



    $$f(x)=3^x-(3-x)$$



    and note that




    • $f(0)=-2$

    • $f(1)=1$


    therefore by IVT there is a solution in $(0,1)$.



    Can you show that this is the unique solution?






    share|cite|improve this answer





















    • If I am correct then: for $x>1$ the term $3^x +x$ is more positive than '3' everywhere in that domain. Also, for x<0 the term $-3+x$ is more negative to be neutralized by $3^x$ as $3^x$ will then become in postive fraction number.
      – jayant98
      13 hours ago






    • 1




      @jayant98 We can consider the sign of $f'(x)=3^x log 3 +1$.
      – gimusi
      13 hours ago








    • 1




      Yes Sir! Another way which is more simple and effective.
      – jayant98
      13 hours ago










    • @jayant98 Call me simply gimusi! Bye
      – gimusi
      13 hours ago













    up vote
    1
    down vote










    up vote
    1
    down vote









    HINT



    Let consider



    $$f(x)=3^x-(3-x)$$



    and note that




    • $f(0)=-2$

    • $f(1)=1$


    therefore by IVT there is a solution in $(0,1)$.



    Can you show that this is the unique solution?






    share|cite|improve this answer












    HINT



    Let consider



    $$f(x)=3^x-(3-x)$$



    and note that




    • $f(0)=-2$

    • $f(1)=1$


    therefore by IVT there is a solution in $(0,1)$.



    Can you show that this is the unique solution?







    share|cite|improve this answer












    share|cite|improve this answer



    share|cite|improve this answer










    answered 14 hours ago









    gimusi

    85.5k74293




    85.5k74293












    • If I am correct then: for $x>1$ the term $3^x +x$ is more positive than '3' everywhere in that domain. Also, for x<0 the term $-3+x$ is more negative to be neutralized by $3^x$ as $3^x$ will then become in postive fraction number.
      – jayant98
      13 hours ago






    • 1




      @jayant98 We can consider the sign of $f'(x)=3^x log 3 +1$.
      – gimusi
      13 hours ago








    • 1




      Yes Sir! Another way which is more simple and effective.
      – jayant98
      13 hours ago










    • @jayant98 Call me simply gimusi! Bye
      – gimusi
      13 hours ago


















    • If I am correct then: for $x>1$ the term $3^x +x$ is more positive than '3' everywhere in that domain. Also, for x<0 the term $-3+x$ is more negative to be neutralized by $3^x$ as $3^x$ will then become in postive fraction number.
      – jayant98
      13 hours ago






    • 1




      @jayant98 We can consider the sign of $f'(x)=3^x log 3 +1$.
      – gimusi
      13 hours ago








    • 1




      Yes Sir! Another way which is more simple and effective.
      – jayant98
      13 hours ago










    • @jayant98 Call me simply gimusi! Bye
      – gimusi
      13 hours ago
















    If I am correct then: for $x>1$ the term $3^x +x$ is more positive than '3' everywhere in that domain. Also, for x<0 the term $-3+x$ is more negative to be neutralized by $3^x$ as $3^x$ will then become in postive fraction number.
    – jayant98
    13 hours ago




    If I am correct then: for $x>1$ the term $3^x +x$ is more positive than '3' everywhere in that domain. Also, for x<0 the term $-3+x$ is more negative to be neutralized by $3^x$ as $3^x$ will then become in postive fraction number.
    – jayant98
    13 hours ago




    1




    1




    @jayant98 We can consider the sign of $f'(x)=3^x log 3 +1$.
    – gimusi
    13 hours ago






    @jayant98 We can consider the sign of $f'(x)=3^x log 3 +1$.
    – gimusi
    13 hours ago






    1




    1




    Yes Sir! Another way which is more simple and effective.
    – jayant98
    13 hours ago




    Yes Sir! Another way which is more simple and effective.
    – jayant98
    13 hours ago












    @jayant98 Call me simply gimusi! Bye
    – gimusi
    13 hours ago




    @jayant98 Call me simply gimusi! Bye
    – gimusi
    13 hours ago










    up vote
    1
    down vote













    Sure. The solution can be expressed in terms of the Lambert W function. In general, graphing is not a way of "solving" an equation because it cannot give you an exact solution. Graphing the equations above will help you approximate $x approx 0.742$ but cannot give you an exact value.






    share|cite|improve this answer

























      up vote
      1
      down vote













      Sure. The solution can be expressed in terms of the Lambert W function. In general, graphing is not a way of "solving" an equation because it cannot give you an exact solution. Graphing the equations above will help you approximate $x approx 0.742$ but cannot give you an exact value.






      share|cite|improve this answer























        up vote
        1
        down vote










        up vote
        1
        down vote









        Sure. The solution can be expressed in terms of the Lambert W function. In general, graphing is not a way of "solving" an equation because it cannot give you an exact solution. Graphing the equations above will help you approximate $x approx 0.742$ but cannot give you an exact value.






        share|cite|improve this answer












        Sure. The solution can be expressed in terms of the Lambert W function. In general, graphing is not a way of "solving" an equation because it cannot give you an exact solution. Graphing the equations above will help you approximate $x approx 0.742$ but cannot give you an exact value.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered 14 hours ago









        Mason

        1,6401325




        1,6401325






















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