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Ln(t) - t = 2; Solve for 't'?

  1. Jul 25, 2010 #1
    1. The problem statement, all variables and given/known data

    Is there a simple way to solve for 't' in the equation: ln(t) - t = 2 ? Just a curiosity. Not for a class. I've browsed to texts that cover logarithmic equations and haven't found a single problem or rule say how to solve for 't'.



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Jul 25, 2010 #2

    rock.freak667

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    Homework Helper

    There is no way to get 't' in terms of elementary functions, but you can solve it graphically or perhaps using another complex function.
     
  4. Jul 25, 2010 #3
    Trancendental equation.
    No closed form solution.
    Turn it into a game.
    Calculate ln(t)-t vs t in Excel.
    Play with t.
    Stop when you get to an answer accurate to 6 decimal places, ie. 2.000001.
    Now you will have developed some feel for the iterative solution of a trancendental equation. You can then read more about Newton-Raphson, etc.
     
  5. Jul 25, 2010 #4

    Mute

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    Homework Helper

    Exponentiate both sides to get:

    [tex]te^{-t} = e^2[/tex]

    Now, multiply both sides by -1 to get [itex]-te^{-t} = -e^2[/itex]. This is now of the form of the Lambert-W function, defined by

    [tex]W(z)e^{W(z)} = z,[/tex]

    which determines t in terms of the lambert W function. This function is not an elementary function, but much is known about it. See http://en.wikipedia.org/wiki/Lambert_W .
     
  6. Jul 25, 2010 #5

    Mathematica 6 found a solution, however the numerical solution has a imaginary axis component.

    ln(t) - t = 2

    t = - ProductLog[-e^2]

    ProductLog[z]
    gives the principal solution for w in z=we^w.

    The ProductLog[z] function is the Lambert-W function, however Mathematica 6 and 7 documentation does not mention this.

    t = -1.13902 - 2.07318i

    Reference:
    http://en.wikipedia.org/wiki/Lambert_W" [Broken]
    http://reference.wolfram.com/mathematica/ref/ProductLog.html" [Broken]
     
    Last edited by a moderator: May 4, 2017
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