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Solution to an exponential equation

  1. Oct 14, 2005 #1
    Today I came across a very strange exponential equation to which neither my teacher nor I could find a solution. It is as follows:
    [itex]e^t=3t^2[/itex]
    This could be easily solved graphically, but could anyone show me how to do this algebraically?
    Thanks in advance!
     
  2. jcsd
  3. Oct 14, 2005 #2

    Tom Mattson

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    I don't think it can be done.
     
  4. Oct 14, 2005 #3
    you would need a second equation describing t. then you could use substitution to solve for it.

    I mean even if you used the natural logarithm you still get:

    [itex]t = 2ln(3t)[/itex] and that does not help you much since you still have t in terms of itself and if you then did this:

    [itex]0 = 2ln(3t) - t[/itex]
    you can not factor out t.

    that does seem to be an interesting solution set though (is there even a solution set?).
     
    Last edited: Oct 14, 2005
  5. Oct 14, 2005 #4

    HallsofIvy

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    In general, an equation involving a a transcendental function (such as et) and an algebraic function (such as t2) can't be solved in terms of "elementary" functions. It could, I think, be solved in terms of the "Lambert W function", which is defined as the inverse function to f(x)= xex.

    (I edited this- my f(x)= xe2 was a typo.)
     
    Last edited: Oct 15, 2005
  6. Oct 14, 2005 #5

    hotvette

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    This reminds me of trying to solve [itex]y = x^x[/itex] for a given y.:smile:
     
  7. Oct 16, 2005 #6

    saltydog

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    The Lambert W-function is defined as the inverse of the following:

    [tex]
    f(x)=xe^x=y
    [/tex]

    then:

    [tex]
    f^{-1}(y)=x=W(y)
    [/tex]

    with W being the Lambert W-function for y>-e^{-1}

    If:

    [tex]
    g(x)=x^2e^x=y
    [/tex]

    then:

    [tex]
    g^{-1}(y)=2W(\frac{\sqrt y}{2})
    [/tex]

    and in general if:

    [tex]
    h(x)=x^ne^x=y
    [/tex]

    then:
    [tex]
    h^{-1}(y)=nW(\frac{y^\frac{1}{n}}{n})
    [/tex]

    Kindly proceed to express the solution of your equation in terms of this generalized Lambda W function, that is:

    [tex]t=\text{some function of W}[/tex]
     
  8. Oct 16, 2005 #7
    Thanks for all for your input!

    I thought I was just missing some easy step...
    guess I was wrong!
     
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