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 November 8th, 2012, 11:19 AM #1 Joined: Nov 2012 Posts: 4 Thanks: 0 e^x-3x=0 What type of solution does e^x-3x=0 have? a finite number, a surd, a fraction, a combination of constants?
 November 8th, 2012, 11:31 AM #2 Global Moderator     Joined: Nov 2006 From: UTC -5 Posts: 12,866 Thanks: 95 Re: e^x-3x=0 I believe you need Lambert's W function for this problem, so I would call it a transcendental solution. There should be two real-valued solutions (and infinitely many complex solutions).
 November 8th, 2012, 11:41 AM #3 Joined: Nov 2012 Posts: 4 Thanks: 0 Re: e^x-3x=0 and can it be written down accurately? for example pi is transcendental, but we can write be instead of 3.14159.... Does the same happen with the solution of x?
November 8th, 2012, 12:15 PM   #4
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Re: e^x-3x=0

Quote:
 Originally Posted by sonicjugi and can it be written down accurately? for example pi is transcendental, but we can write be instead of 3.14159.... Does the same happen with the solution of x?
I dont understand, what do you mean by accurately?

 November 8th, 2012, 12:17 PM #5 Global Moderator     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 11,566 Thanks: 108 Math Focus: The calculus Re: e^x-3x=0 You can find as many digits as you wish by using a root finding technique such as Newton's method.
November 8th, 2012, 02:32 PM   #6
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Re: e^x-3x=0

Quote:
 Originally Posted by MarkFL You can find as many digits as you wish by using a root finding technique such as Newton's method.
Yes. In this case higher-order techniques will have better convergence properties, I think. I use a third-order technique in my program which calculates the W-function, for example.

 November 11th, 2012, 08:41 AM #7 Joined: Oct 2012 Posts: 22 Thanks: 0 Re: e^x-3x=0 e^x-3x=0 e^x=3x e^-x=(1/3x) powering both sides to (1/3x) e^(-x * (1/3x))=(1/3x)^(1/3x) e^(-1/3)=(1/3x)^(1/3x) then 1/3x = ssrt(e^(-1/3)) , ssrt is super sequare root used in tetration. ssrt (z) = 1/(z^^n) when n goes to infinity then x = 1/(3*(ssrt(e^(-1/3))))
 November 11th, 2012, 08:44 AM #8 Joined: Oct 2012 Posts: 22 Thanks: 0 Re: e^x-3x=0 correction ssrt (z) = 1/((1/z)^^n) when n goes to infinity
November 11th, 2012, 10:33 AM   #9
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Re: e^x-3x=0

Quote:
 Originally Posted by nsrmsm ssrt is super sequare root used in tetration.
Right, this is essentially equivalent to Lambert's W function in terms of expression power. ssrt(x) = log(x)/W(log(x)) ~ log x/log log x as x increases without bound.

November 11th, 2012, 02:24 PM   #10

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Re: e^x-3x=0

Quote:
Originally Posted by CRGreathouse
Quote:
 Originally Posted by nsrmsm ssrt is super sequare root used in tetration.
Right, this is essentially equivalent to Lambert's W function in terms of expression power. ssrt(x) = log(x)/W(log(x)) ~ log x/log log x as x increases without bound.
Lambert's W function is not elementry function

but ssrt(x) is elementry tool because looks like srt(x).

that is why I prefer ssrt(x) than W(x).

 November 11th, 2012, 04:22 PM #11 Global Moderator     Joined: Nov 2006 From: UTC -5 Posts: 12,866 Thanks: 95 Re: e^x-3x=0 Neither ssrt nor W are elementary functions.
November 11th, 2012, 10:26 PM   #12

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Re: e^x-3x=0

Quote:
 Originally Posted by CRGreathouse Neither ssrt nor W are elementary functions.
It shoud be

elementry functions should be enhanced by tetration.
because they can not keep up with modern sciences with the old tools
so if we call this f(X) = srt(x) elementry

why do not call this f(x) = ssrt(x) also elementry

while they use the same method that they are roots of x*x or x^x.

 November 11th, 2012, 10:47 PM #13 Global Moderator     Joined: Nov 2006 From: UTC -5 Posts: 12,866 Thanks: 95 Re: e^x-3x=0 If you call that function elementary, you'll have to call W elementary too.
 November 12th, 2012, 01:31 AM #14 Math Team     Joined: Mar 2012 From: India, West Bengal Posts: 3,815 Thanks: 42 Math Focus: Number Theory Re: e^x-3x=0 Tetration is not considered an elementary one even though addition multiplication and exponentiation are considered elemntary. However, one can prove that $\int \frac{dx}{{}^a x}$ isn't elementary in [x, ln(x)] and proving the non-elementaryness of tetration. However, the truth of this theorem and the converse are still some lose conjectures of mine : viewtopic.php?f=22&t=32937
November 16th, 2012, 07:18 AM   #15

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Re: e^x-3x=0

Quote:
 Originally Posted by CRGreathouse If you call that function elementary, you'll have to call W elementary too.
look
if y = x^2 then x = srt(y),
so if y = x^x then x = ssrt(y)

Now
if y = x + ex then x = y/(1+e),
so if y = xe^x you can not say that x = W(y) this is not elementry
but you can say x = y/(ssrt(e^y))

note that
x [1] e[2]x is equivelent (in hyperoperations) to x [2] e[3]x

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