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Homework Help: Logarithm question, finding all possible pairs of integers

  1. Dec 3, 2012 #1
    1. The problem statement, all variables and given/known data
    Find all possible pairs of integers a and n such that:

    log(1/n)(√(a+√(15)) - √(a -√(15)))=-1/2

    (that's log to the base (1/n))


    3. The attempt at a solution

    (1/n)^-1/2 = (√(a+√(15)) - √(a -√(15))
    ∴ n^4 = (a+√(15) - (a -√(15) - 2√((a+√15)(a -√(15))
    ∴ n^4 = =2√(15) - 2√((a+√15)(a -√(15))
    eventually simplifying to:
    n^(16)-√(15)n^4 =4a^2

    dont know how to solve, probably made mistake

    question is from core 3 edexcel and is worth 13 marks
     
  2. jcsd
  3. Dec 3, 2012 #2
  4. Dec 3, 2012 #3

    symbolipoint

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    Typing in all the steps into the Compose form would be a mess; but one of my partial results seems to give the equation,
    (n-2a)/(-2) = sqrt(a2-15)
     
  5. Dec 3, 2012 #4
    Yeah, I get the same thing.
     
  6. Dec 3, 2012 #5

    symbolipoint

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    Further steps give me n2-4an+60=0, from which solving for a,
    a=(n/4)+(15/n).

    That could let us find possible pairs of INTEGERS for a and n.
     
  7. Dec 3, 2012 #6
    OK, but maybe we should let the OP solve it? :tongue:
     
  8. Dec 3, 2012 #7

    symbolipoint

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    Knowing no clever way to solve specifically for integer solutions, would a BASIC FOR loop be acceptable? I would run n from about 0.100 to 50, incrementing by 0.100 for each step. a would be calculated in each run through the loop.

    ( I know no clever, fancy way to find the integer solutions for this rational equation but I believe a BASIC program can expose some integer number pairs for n and a ).
     
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