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  1. #1
    dan
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    system

    can i solve the system

    x^n = 12
    n^x = 4

    i keep running into problems like n=1 and x= (LN(4))/0
    (wew... don't let the perfecthacker see that)

    or would the values just be undifind??

    dan
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  2. #2
    MHF Contributor Quick's Avatar
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    Quote Originally Posted by dan View Post
    can i solve the system

    x^n = 12
    n^x = 4

    i keep running into problems like n=1 and x= (LN(4))/0
    (wew... don't let the perfecthacker see that)

    or would the values just be undifind??

    dan
    x=\sqrt[n]{12}

    x=ln(4)/ln(n)

    thus: ln(4)/ln(n)=\sqrt[n]{12}

    then: ln(4)=ln(n)*\sqrt[n]{12}

    therefore: ln(4)=ln(n^(12^1/n))

    thus: 4=n^(12^1/n)

    and then I'm stuck
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  3. #3
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by dan View Post
    can i solve the system

    x^n = 12
    n^x = 4

    i keep running into problems like n=1 and x= (LN(4))/0
    (wew... don't let the perfecthacker see that)

    or would the values just be undifind??

    dan
    The best I can do for you is this:
    From the second equation, if we take ln of both sides:
    ln(n^x) = ln(4)

    x ln(n) = ln(4)

    x = ln(4)/ln(n)

    Insert this into the first equation:

    [ln(4)/ln(n)]^n = 12

    This is, at least, an equation in one variable. Supposedly you can solve this numerically.

    -Dan
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  4. #4
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    I belive you can use Lambert function on,
    (though I did not try it)
    Attached Thumbnails Attached Thumbnails system-picture4.gif  
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  5. #5
    Grand Panjandrum
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    Quote Originally Posted by ThePerfectHacker View Post
    I belive you can use Lambert function on,
    (though I did not try it)
    How does the last line follow from the one before?
    Attached Thumbnails Attached Thumbnails system-gash.jpg  
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  6. #6
    Grand Panjandrum
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    Quote Originally Posted by dan View Post
    can i solve the system

    x^n = 12
    n^x = 4

    i keep running into problems like n=1 and x= (LN(4))/0
    (wew... don't let the perfecthacker see that)

    or would the values just be undifind??

    dan
    This can be done numerically, when one finds:

    x~=8.1272946
    n~=1.1859838

    RonL
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  7. #7
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    Quote Originally Posted by CaptainBlack View Post
    How does the last line follow from the one before?
    A mistake.
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  8. #8
    dan
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    Quote Originally Posted by CaptainBlack View Post
    This can be done numerically, when one finds:

    x~=8.1272946
    n~=1.1859838

    RonL

    hmmm thanks did you do that with a computer progam??

    dan
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  9. #9
    Grand Panjandrum
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    Quote Originally Posted by dan View Post
    hmmm thanks did you do that with a computer progam??

    dan
    Yes, but I was even lazier than normal, what I did was use the Excel solver
    to minimise:

    f(x,n)=(x^n-12)^2+(n^x-4)^2,

    which since this is always >=0, has minima at the roots of the system.

    RonL
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  10. #10
    dan
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    Quote Originally Posted by CaptainBlack View Post
    Yes, but I was even lazier than normal, what I did was use the Excel solver
    to minimise:

    f(x,n)=(x^n-12)^2+(n^x-4)^2,

    which since this is always >=0, has minima at the roots of the system.

    RonL
    cool...those are probebly illagle for students....^_^
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