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Solve for x (3^x = x^2)

  1. Dec 27, 2011 #1
    1. The problem statement, all variables and given/known data

    Find all possible values of x:

    x2 = 3x

    2. Relevant equations



    3. The attempt at a solution

    I took the log of both sides.. didn't get anywhere. I'm not sure where to start.
     
  2. jcsd
  3. Dec 27, 2011 #2

    micromass

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    You can't solve this analytically with logarithms. You can either guess the right answer and prove that it works, use the Lambert W function or you can use a numerical method.
     
  4. Dec 27, 2011 #3
    okays... thanks a lot!
     
  5. Dec 27, 2011 #4

    dextercioby

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    Your equation has a solution, because both functions are monotone and increasing.

    [tex] 3^{-1}<(-1)^2 [/tex]

    [tex] 3^0 > 0^2 [/tex]
     
  6. Dec 27, 2011 #5

    D H

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    One simple way to find that solution is via a fixed point iteration. Taking the square root of both sides and recognizing that the solution is negative yields
    [tex]x=-\sqrt{3^x}[/tex]
    This converges to the solution fairly quickly. Quickly by fixed point iteration standards, that is. Fixed point iteration is rarely fast.
     
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