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Math Help - solving 'exponential' equation

  1. #1
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    solving 'exponential' equation

    Hi:


    Is there an algebraic means by which to solve this equation and others like it? I can estimate via Newton's method but I would like to know how to determine solution(s) exactly.


    e^x + 2x = 7


    Thank you.


    Rich B.
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by Rich B. View Post
    Hi:


    Is there an algebraic means by which to solve this equation and others like it? I can estimate via Newton's method but I would like to know how to determine solution(s) exactly.


    e^x + 2x = 7


    Thank you.


    Rich B.
    This can be solved using Lambert's W function:

    Let u = 7-2x, then x = (7-u)/2, so the equation becomes:

    e^x = 7 - 2x,

    e^{7/2} e^{-u/2} = u

    e^{7/2} = u e^{u/2}

    (1/2) e^{7/2} = (u/2) e^{u/2}

    So u = 2 W((1/2) e^{7/2}) = 4.15255, and x= 1.42372



    RonL
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  3. #3
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    I developed a way to solve,
    ax+b=e^x if a!=0.

    Here.

    When my engineering professor boldy announced that there is no way to solve this in closed form. I wanted to show to him that he was wrong. (I love doing that).
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  4. #4
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    Hi Ron:

    I followed you right up to the last line. I don't know what W(...) means. Is that 'function W of the argument 0.5e^3.5'? And if so, what is the function W? Confused...

    I will be away for a week, effective Saturday, A.M., so if I don't get right back with the proper 'thanks for filling in the gaps', you will understand I trust.

    P.Hckr: Thank you for your response. I gave your work a quick read upon getting in at 1:30 A.M., and I will definitely need a good night's sleep before re-reading and digesting.

    Thank you both,

    Rich B.
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  5. #5
    Grand Panjandrum
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    Quote Originally Posted by Rich B. View Post
    Hi Ron:

    I followed you right up to the last line. I don't know what W(...) means. Is that 'function W of the argument 0.5e^3.5'? And if so, what is the function W? Confused...

    I will be away for a week, effective Saturday, A.M., so if I don't get right back with the proper 'thanks for filling in the gaps', you will understand I trust.

    P.Hckr: Thank you for your response. I gave your work a quick read upon getting in at 1:30 A.M., and I will definitely need a good night's sleep before re-reading and digesting.

    Thank you both,

    Rich B.
    W is Lambert's W function. W(z) is the inverse function of:

    f(w)=w exp(w)

    So if z=w exp(w), then w=W(z). It is considered by a number of people that
    it ought to be included in the list of elementary transcendental functions.

    You can learn more about it here.

    That W cannot be written as a finite combination of the other elementary
    functions and algebraic operations (as far as I know anyway) shows that
    a general equation of the form:

    exp(x) + ax + b = 0

    cannot be solved using just the normal elemantary function and algebraic
    operations in a finite closed form.

    RonL
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  6. #6
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    Quote Originally Posted by CaptainBlank View Post
    That W cannot be written as a finite combination of the other elementary
    functions and algebraic operations (as far as I know anyway)
    From what I hear as well.
    I cannot imagine how it is proved.

    But do not worry, in several more years I shall know everything there is to know in math, and I will tell you how it is done.
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  7. #7
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by ThePerfectHacker View Post
    But do not worry, in several more years I shall know everything there is to know in math
    What confidence!

    If it's possible i hope you attain it, i hope i attain it as well--though it will be significantly harder for me than for you
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