How to figure out answer to logarithm


by mycrafish
Tags: figure, logarithm
mycrafish
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#1
Mar17-12, 07:41 AM
P: 4
x*log(x)=0.1*x^2
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scurty
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#2
Mar17-12, 08:07 AM
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My guess is that using Newton's Method would be the easiest. I don't see any easy way to isolate x in that equation.
NascentOxygen
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Mar17-12, 05:57 PM
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Quote Quote by mycrafish View Post
x*log(x)=0.1*x^2
Is log = log10? Then I'd guess about, oh, let's see, .... 10?

HallsofIvy
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Mar17-12, 10:37 PM
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How to figure out answer to logarithm


First, of course, x= 0 is not a solution because log(0) is not defined. So you can divide both sides by x to get log(x)= 0.1 x. You can now write this as [itex]x= e^{0.1x}[/itex]. If you let y= -0.1x, that becomes [itex]-y/0.1= e^{-y}[/itex]. Now multiply on both sides by [itex]-0.1e^y[/itex] to get [itex]ye^y= -0.1[/itex].

Now we can take the Lambert W function of both sides:
y= -0.1x= W(-0.1) so x= -10W(-0.1)= -10(-0.111833)= 1.11833 (to six significant figures).

(The Lambert W function is defined as the inverse function to [itex]f(x)= xe^x[/itex]. It is also known as the "ProductLog" function. Mathematica evaluates that function and it can be evaluated at http://functions.wolfram.com/webMath...ame=ProductLog. That's what I used to get the value above.)


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