November 8th, 2012, 11:19 AM  #1 
Joined: Nov 2012 Posts: 4 Thanks: 0  e^x3x=0
What type of solution does e^x3x=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: 12,864 Thanks: 95  Re: e^x3x=0
I believe you need Lambert's W function for this problem, so I would call it a transcendental solution. There should be two realvalued solutions (and infinitely many complex solutions).

November 8th, 2012, 11:41 AM  #3 
Joined: Nov 2012 Posts: 4 Thanks: 0  Re: e^x3x=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  
Math Team Joined: Mar 2012 From: India, West Bengal Posts: 3,815 Thanks: 42 Math Focus: Number Theory  Re: e^x3x=0 Quote:
 
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^x3x=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  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 12,864 Thanks: 95  Re: e^x3x=0 Quote:
 
November 11th, 2012, 08:41 AM  #7 
Joined: Oct 2012 Posts: 22 Thanks: 0  Re: e^x3x=0
e^x3x=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^x3x=0
correction ssrt (z) = 1/((1/z)^^n) when n goes to infinity 
November 11th, 2012, 10:33 AM  #9  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 12,864 Thanks: 95  Re: e^x3x=0 Quote:
 
November 11th, 2012, 02:24 PM  #10  
Joined: Oct 2012 Posts: 22 Thanks: 0  Re: e^x3x=0 Quote:
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: 12,864 Thanks: 95  Re: e^x3x=0
Neither ssrt nor W are elementary functions.

November 11th, 2012, 10:26 PM  #12  
Joined: Oct 2012 Posts: 22 Thanks: 0  Re: e^x3x=0 Quote:
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: 12,864 Thanks: 95  Re: e^x3x=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^x3x=0
Tetration is not considered an elementary one even though addition multiplication and exponentiation are considered elemntary. However, one can prove that isn't elementary in [x, ln(x)] and proving the nonelementaryness 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  
Joined: Oct 2012 Posts: 22 Thanks: 0  Re: e^x3x=0 Quote:
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  