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FoolForMath

  • 3 years ago

Easy yet cute problem, if \( p^{12}=q^6=r^3=s^2, p \neq 1 \). Find the value of \(\log_p pqrs \). Lets see who can solve this the fastest. Pleas read on for the extra question (posted later): How many values of \(x \) satisfy the equation \( \log (2x)=\frac 14 \log (x-15)^4 \)?

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  1. matricked
    • 3 years ago
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    is it 13

  2. apoorvk
    • 3 years ago
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    23/12

  3. apoorvk
    • 3 years ago
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    13

  4. Limitless
    • 3 years ago
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    q=p^2 r=p^4 s=p^6 2+4+6+1=13

  5. apoorvk
    • 3 years ago
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    I think too much actually :/ and act too hastily. :/ It is 13.

  6. FoolForMath
    • 3 years ago
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    I like posting sitters for a change ... :P

  7. asnaseer
    • 3 years ago
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    I think it can have several values:\[1+4\pm2\pm6\]

  8. asnaseer
    • 3 years ago
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    since q and s can be negative

  9. Limitless
    • 3 years ago
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    Asnaseer, we do not know if the even number radical function is defined for strictly positive or negative. Positive is assumed.

  10. asnaseer
    • 3 years ago
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    so are you saying, for example, that:\[s=-p^6\]is not a valid solution to this?

  11. Limitless
    • 3 years ago
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    Hmm... Rather hard to say exactly what I mean. It's true that \(s=-p^6\). But, I assumed we were using the positive roots only because that's conventional. Otherwise, as you see, our function is actually a multivalued relation rather than a function and thus the logarithm becomes murkily defined. (I am potentially abusing terminology here.)

  12. asnaseer
    • 3 years ago
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    maybe FFM can clarify?

  13. Limitless
    • 3 years ago
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    My main point is that if we allow ourselves to take both of the roots, the logarithm is no longer a function. This can't be the case, since the logarithm is always a function.

  14. asnaseer
    • 3 years ago
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    BTW: You are most probably correct Limitless. I am not a /true/ mathematician as such - I only follow maths as a hobby.

  15. Limitless
    • 3 years ago
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    I hope you realize that I am still impressed by your solution, critique aside.

  16. asnaseer
    • 3 years ago
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    :) thx

  17. FoolForMath
    • 3 years ago
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    Asnaseer is technically right, but I am happy with the less pedantic solution for this one.

  18. asnaseer
    • 3 years ago
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    Is there some article on this area that I could read about to help me understand this better?

  19. Limitless
    • 3 years ago
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    asnaseer, precisely what confuses you? This is more of a notion of convention rather than of serious mathematical concepts.

  20. asnaseer
    • 3 years ago
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    your description of log being a function and therefore it can only have one value - I was just wondering if there was any formal definition of this concept somewhere?

  21. apoorvk
    • 3 years ago
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    Well. I seem to think that 'q' and 's' can be negative, but their resultant powers would definitely be positive - so how does the question of the powers being 'negative' arise? s= (-p)^6 ---> alrighty! but the exponent can only be '6' right? of the exponent changes sign, then the whole thingy changes. Or am I totally wrong once again? -_-

  22. Limitless
    • 3 years ago
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    http://en.wikipedia.org/wiki/Function_(mathematics)#Formal_definition :) I rather love formal definitions if they are also intuitively beautiful. This is one such case. The reason that functions are considered injective and surjective (i.e. one-to-one, which is what I have been saying) is that this allows us to suppose the existence of their inversion via another function! It's rather interesting stuff. :)

  23. asnaseer
    • 3 years ago
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    good point apoorvk!

  24. asnaseer
    • 3 years ago
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    does that mean that:\[\log_p(-p)^6\]is undefined?

  25. asnaseer
    • 3 years ago
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    thx for the link Limitless.

  26. asnaseer
    • 3 years ago
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    sorry I meant:\[\log_p-(p)^6\]

  27. FoolForMath
    • 3 years ago
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    In real yes.

  28. Limitless
    • 3 years ago
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    hmm.. Good question. \(\log_{p}(-p^6)=\log_{p}(-1)+\log_{p}(p^6)=\log_{p}(-1)+6\) Here's an intuitive way of thinking: What in the world is \(\log_{p}(-1)\)???

  29. FoolForMath
    • 3 years ago
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    I mean it is not defined in \( \mathbb{R}\)

  30. asnaseer
    • 3 years ago
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    It always amazing me how even /seemingly/ simple problems can generate a whole raft of new learning! :)

  31. asnaseer
    • 3 years ago
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    *amazes

  32. apoorvk
    • 3 years ago
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    Yup^!

  33. FoolForMath
    • 3 years ago
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    There is a tricky problem on this topic that I encountered last month,I will post it some time later.

  34. asnaseer
    • 3 years ago
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    BTW: The wolf says \(\log(-1)=i\pi\)

  35. asnaseer
    • 3 years ago
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    so FFM - you are right - it does not belong in the reals

  36. Limitless
    • 3 years ago
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    You will find it very intriguing if you ask yourself why we restrict ourselves to certain domains on popular function. It taught me many deep ideas to ponder this. A rather elementary reason is that much of a function cannot even be graphed on a real 2-dimensional plane. Consider \(f(x)=x^2\) at \(x=i\). It is clear that we can graph \(f(i)=-1\), however how do we plot \(i\)??? This is where it becomes very intriguing: Complex numbers require special plotting techniques or special geometric spaces. :)

  37. asnaseer
    • 3 years ago
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    You just melted by brain Limitless! (in a good way) :D

  38. Limitless
    • 3 years ago
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    Well, I am glad I could! :) May someone more enlightened than me correct me if I am misrepresenting things.

  39. FoolForMath
    • 3 years ago
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    How many values of \(x\) satisfy the equation \( \log (2x)=\frac 14 (x-15)^4 \)?

  40. lgbasallote
    • 3 years ago
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    42 =_=

  41. asnaseer
    • 3 years ago
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    Limitless: That little plotting problem you posed up there is indeed very intriguing. Looks like my weekend is going to be filled with reading material :)

  42. Limitless
    • 3 years ago
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    Good lord, FFM. Surely this does not require numerical methods. D: I hope so, Asn. I had much spare time in class, LOL. You ask yourself questions like this out of boredom.

  43. FoolForMath
    • 3 years ago
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    Well for complex numbers we need a completely new reference axes, The argand plane.

  44. FoolForMath
    • 3 years ago
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    *plotting

  45. asnaseer
    • 3 years ago
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    FFM: Can this be solved analytically?

  46. FoolForMath
    • 3 years ago
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    Oh yes, 1 line if you start sufficiently from left ;)

  47. asnaseer
    • 3 years ago
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    ah! - two values?

  48. Limitless
    • 3 years ago
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    This whole factor \((x-15)^5\) is turning me off.

  49. FoolForMath
    • 3 years ago
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    No. Btw consider \(\mathbb{R} \)

  50. asnaseer
    • 3 years ago
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    I was thinking in \(\log(2x)\), 2x cannot be negative, therefore x must be positive. And I /assumed/ that \(\frac 14 (x-15)^4\) would have 2 negative and 2 positive solutions, so only 2 solutions possible.

  51. Limitless
    • 3 years ago
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    \(x\) is not an integer. Is this true?

  52. FoolForMath
    • 3 years ago
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    No.

  53. experimentX
    • 3 years ago
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    \( \log(2x)^4 = (x-15)^4\)

  54. Limitless
    • 3 years ago
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    I missed something, then.

  55. experimentX
    • 3 years ago
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    |dw:1338633080637:dw|

  56. experimentX
    • 3 years ago
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    can't find the solution analytically though!!

  57. asnaseer
    • 3 years ago
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    that graph makes it look like there are indeed only 2 solutions

  58. Limitless
    • 3 years ago
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    Which are symmetric.

  59. asnaseer
    • 3 years ago
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    symmetric in which sense?

  60. experimentX
    • 3 years ago
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    yes ... expoential function >> polynomial function >> log function

  61. Limitless
    • 3 years ago
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    Oops. I was wrong, sorry.

  62. FoolForMath
    • 3 years ago
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    Then the two solutions are x =? and x = ? .. :)

  63. experimentX
    • 3 years ago
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    lol ... i hope you don't mind if i use guessing technique

  64. asnaseer
    • 3 years ago
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    FFM: you are changing the goal posts here - your question was: How many values of x satisfy the equation...

  65. FoolForMath
    • 3 years ago
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    Yes, but can't we find the solutions if there are two solutions?

  66. experimentX
    • 3 years ago
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    i asked the same type of question of MSE they showed me Newton's method

  67. experimentX
    • 3 years ago
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    lol :D

  68. Limitless
    • 3 years ago
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    ^yes.

  69. Limitless
    • 3 years ago
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    Exact same experience. I don't even think Lambert W can be applied.

  70. asnaseer
    • 3 years ago
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    is the log using base 10 or base e?

  71. FoolForMath
    • 3 years ago
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    common logarithm base 10 I would say.

  72. FoolForMath
    • 3 years ago
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    Well x=5 is the only solution. If you find the other one please let me know :)

  73. Limitless
    • 3 years ago
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    \(x=5\) isn't a solution. Unless my math skills are horribly wrong right now.

  74. asnaseer
    • 3 years ago
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    x=5 does not seem to satisfy the equation?

  75. asnaseer
    • 3 years ago
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    this is the equation right?:\[\log (2x)=\frac 14 (x-15)^4\]

  76. FoolForMath
    • 3 years ago
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    I just checked in mathematica: x = 5; 1/4 Log[10, (x - 15)^4] = Log [10, 2 x]=1

  77. FoolForMath
    • 3 years ago
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    OMG!! I didn't mention the log carp :(

  78. experimentX
    • 3 years ago
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    what am i getting?? http://www.wolframalpha.com/input/?i=solve+%28x-15%29%5E4+-+log%5B10%2C+%282x%29%5E4%5D+%3D+0

  79. FoolForMath
    • 3 years ago
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    \[ \log (2x)=\frac 14 \log (x-15)^4 \]

  80. asnaseer
    • 3 years ago
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    bad FFM - bad FFM - go stand in a corner now! :)

  81. FoolForMath
    • 3 years ago
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    VERY VERY VERY poor penman ship :( My apologies to all :(

  82. Limitless
    • 3 years ago
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    Whoa whoa whoa. One moment. Is this \(\log(x-15)^4\) or is it meant to be \(\log[(x-15)^4]\)?

  83. FoolForMath
    • 3 years ago
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    I think both are same.

  84. Limitless
    • 3 years ago
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    That implies \(x=-15\) is a solution.

  85. FoolForMath
    • 3 years ago
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    No. that's the tricky part. Try substituting x=-15 in the equation.

  86. Limitless
    • 3 years ago
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    So, probably not. But this is cool, so here you go: \(\log(2x)=\frac{1}{4}\log((x-15)^4)\) \(4\log(2x)=\log((x-15)^4)\) \(\log((2x)^4)=\log((x-15)^4)\) \(2x=x-15\) \(x=-15\)

  87. asnaseer
    • 3 years ago
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    \[\log(2x)=\frac{1}{4}\log(x-15)^4\]therefore:\[\log(2x)^4=\log(x-15)^4\]\[(2x)^4=(x-15)^4\]\[2x=\pm(x-15)\]

  88. FoolForMath
    • 3 years ago
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    Yes yes but as I said x=-15 is not a valid solution. Okay I am going to going to dig a hole and hide my face on it. I will see you guys later.

  89. asnaseer
    • 3 years ago
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    therefore only ONE real solution, x=5

  90. FoolForMath
    • 3 years ago
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    That's right asnaseer.!

  91. Limitless
    • 3 years ago
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    Asnaseer has slain me. :p

  92. Limitless
    • 3 years ago
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    I forgot, once again, the \(\pm\).

  93. asnaseer
    • 3 years ago
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    :) we all have our days - in my case - seconds, of fame :)

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