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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
    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


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    Science Advisor
    Homework Helper

    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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    Staff Emeritus
    Science Advisor

    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
    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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