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.

Quote:

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Originally Posted by Rich B.

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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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

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).

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.

Quote:

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Originally Posted by Rich B.

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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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

Quote:

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Originally Posted by CaptainBlank

That W cannot be written as a finite combination of the other elementary
functions and algebraic operations (as far as I know anyway)

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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.

Quote:

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Originally Posted by ThePerfectHacker

But do not worry, in several more years I shall know everything there is to know in math

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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