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Homework Help: Simple Indices

  1. May 4, 2010 #1
    1. The problem statement, all variables and given/known data
    2^x+3^x=13, x=2. How do i prove it?

    3. The attempt at a solution
    i did this
    x log 2 + x log 3 = log 13
    x(log 2 + log 3)=log 13
    x(0.301+0.477)= 1.11
    0.778x=1.11
    x=1.43
     
  2. jcsd
  3. May 4, 2010 #2

    HallsofIvy

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    ??? Why not just substitute "2" for "x"?

    [itex]2^x+ 3^x= 2^2+ 3^2= 4+ 9= 13[/itex].

    You are not asked so solve the equation!
     
  4. May 4, 2010 #3

    tiny-tim

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    Hi Champdx! :smile:

    (try using the X2 tag just above the Reply box :wink:)
    Nooo :redface:

    that's log (x2x3) :wink:
     
  5. May 4, 2010 #4
    Actually i know that the answer x=2 but how am i suppose to prove it?
     
  6. May 4, 2010 #5

    tiny-tim

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    If the question says "solve 2x + 3x = 13", then there's no exact way of doing it, you'll have to use an approximation method (or guess).

    If the question says "show that 2x + 3x = 13 has a solution x = 2", or "prove that 2x + 3x = 13 has a solution x = 2", then all you need to do is to show that 22 + 32 = 13. :smile:
     
  7. May 4, 2010 #6

    Mark44

    Staff: Mentor

    You can't get the equation above from the one you started with. log(A + B) [itex]\neq[/itex] logA + logB. What you did was to take the log of both sides to get
    log(2^x + 3^x) = log 13. That's a legitimate step, but it doesn't lead you anywhere.
    The problem is that log(2^x + 3^x) [itex]\neq[/itex] log 2^x + log 3^x.
     
  8. May 5, 2010 #7
    Is there any way to solve this question by calculation?

    Thanks.
     
  9. May 5, 2010 #8

    tiny-tim

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    Nope. :redface:
     
  10. May 5, 2010 #9
    Champ, read this quote again. There are only two options:
    1) "Solve..." or
    2) "Show that"/"Prove..."

    There's a phrase that comes up frequently in math (that honestly kinda p*sses me off!)...
    "By inspection...".
    This means "I can SEE the answer, but I can't/won't show HOW you could get the same answer without guessing". There might be a better description out there for this phrase, which I think applies to our situation.

    We can see that x = 2 is a solution. We can see that it is a UNIQUE solution because increasing/decreasing x will increase/decrease BOTH terms on the left hand side of the equation...
     
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