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Math Help - Solution to equation

  1. #1
    MHF Contributor Mathstud28's Avatar
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    Solution to equation

    I was in the shower and I was just thinking is the solution to 1-\log_a(x)=-\log_x(a)

    x=a^{\phi}?
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  2. #2
    Moo
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    Yop,

    What is \phi?
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  3. #3
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by Moo View Post
    Yop,

    What is \phi?
    The golden ratio \phi=\frac{1+\sqrt{5}}{2}
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  4. #4
    Moo
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    Well, I find the same thing yep ~

    I can't work with \log_a :

    1-\frac{\ln(x)}{\ln(a)}=-\frac{\ln(a)}{\ln(x)}

    \ln(a)\ln(x)-\ln^2(x)=-\ln^2(a)

    Dividing by \ln^2(a) :

    (\frac{\ln(x)}{\ln(a)})^2-\frac{\ln(x)}{\ln(a)}-1=0

    Hence x=a^\varphi

    Is it what you were thinking about ?



    PS : I rather prefer \varphi than \phi
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  5. #5
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by Moo View Post
    Well, I find the same thing yep ~

    I can't work with \log_a :

    1-\frac{\ln(x)}{\ln(a)}=-\frac{\ln(a)}{\ln(x)}

    \ln(a)\ln(x)-\ln^2(x)=-\ln^2(a)

    Dividing by \ln^2(a) :

    (\frac{\ln(x)}{\ln(a)})^2-\frac{\ln(x)}{\ln(a)}-1=0

    Hence x=a^\varphi




    PS : I rather prefer \varphi than \phi
    Isnt that cool...I did it a different way 1-\log_a(x)=-\log_x(a)\Rightarrow{1-\log_a(x)=\frac{-1}{log_a(x)}}...then letting \log_a(x)=u and multiplying through and using quadratic formula I came up with \phi...thats right \phi
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  6. #6
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by Mathstud28 View Post
    Isnt that cool...I did it a different way 1-\log_a(x)=-\log_x(a)\Rightarrow{1-\log_a(x)=\frac{-1}{log_a(x)}}...then letting \log_a(x)=u and multiplying through and using quadratic formula I came up with \phi...thats right \phi
    this is the way i'd do it
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  7. #7
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    Interesting..I wonder why I've never seen that before..

    I'm just relieved to find out that I'm not the only one who thinks about math in the shower.
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  8. #8
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by elizsimca View Post
    Interesting..I wonder why I've never seen that before..
    yes, it is interesting.

    How did you come up with the question Mathstud?

    I'm just relieved to find out that I'm not the only one who thinks about math in the shower.
    nope, you're the only one. Mathstud was joking when he said that...
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