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November 8th, 2012, 11:19 AM   #1
 
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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?
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November 8th, 2012, 11:31 AM   #2
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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).
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November 8th, 2012, 11:41 AM   #3
 
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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?
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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?
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November 8th, 2012, 12:17 PM   #5
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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.
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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.
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November 11th, 2012, 08:41 AM   #7
 
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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))))
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November 11th, 2012, 08:44 AM   #8
 
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Re: e^x-3x=0

correction

ssrt (z) = 1/((1/z)^^n) when n goes to infinity
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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.
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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).
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