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Solving for x in logarithm problem

  1. Aug 11, 2011 #1
    1. The problem statement, all variables and given/known data
    The equation (attached as image) has
    (a)one irrational solution
    (b)no prime solution
    (c)two real solutions
    (d)one integral solution

    i would like to get help on how to find the possible values of x

    2. Relevant equations
    ( the equation is attached below)


    3. The attempt at a solution
    i solved the equation by applying the base changing property and then doing
    L.C.M of both equations but after that i am unable to solve further?

    Any help will be highly appreciated.
     

    Attached Files:

  2. jcsd
  3. Aug 11, 2011 #2

    SammyS

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    Show what you tried so far.

    What results did you have?

    I suggest: Do change of base with base of 2 .
     
  4. Aug 11, 2011 #3
    The equation (attached as image) has
    (a)one irrational solution
    (b)no prime solution
    (c)two real solutions
    (d)one integral solution

    i would like to get help on how to find the possible values of x

    i solved the equation by applying the base changing property and then doing
    L.C.M of both equations but after that i am unable to solve further?

    Any help will be highly appreciated.
     

    Attached Files:

  5. Aug 11, 2011 #4
    Because the numbers seem to be related to powers of 2, I was convinced to work with logarithm base 2. Let [itex]lb[/itex] stand for [itex]log_{2}[/itex].

    We then have

    [tex]log_{x^2}(16) + log_{2x}(64)[/tex]
    [tex]= \frac{lb(16)}{2\cdot lb(x)} + \frac{lb(64)}{lb(2x)}[/tex]
    [tex]= \frac{2}{lb(x)} + \frac{6}{lb(x) + 1} = 3[/tex]

    this reduces to a quadratic which should be easily solvable.
     
  6. Aug 12, 2011 #5
    2/log2(x) + 6/ (log2(x)+1)=3
    i got this result
    after that what should i do
     
  7. Aug 12, 2011 #6
    logx^2 16 + log2x 64 = 3
    log(16) /log (x2) + log(64) /log(2x) = 3
    log(2x) log(16) + log(64) log (x2) = 3 log(2x) log (x2)
    log(2x) log(24) + log(26) log (x2) = 3 log(2x) log (x2)
    4 (log 2+log x) log(2) + 12 log(2) log (x) = 6 (log 2 + log x) log x
    4 (log 2)2 + 4 log (x) log (2) + 12 log (2) log (x) = 6 log (2) log (x) + 6 (log x)2
    4 (log 2)2 + 10 log (2) log (x) = 6 (log x)2

    Solve the quadratic formula for log x and you'll get your answer.
     
    Last edited: Aug 12, 2011
  8. Aug 12, 2011 #7
    :smile:Thanks a lot i have got the answer:smile:
    answer are x=4 and x=1[itex]/\sqrt[3]{}2[/itex]
     
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