How to figure out answer to logarithm

  • Thread starter mycrafish
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  • #1
mycrafish

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x*log(x)=0.1*x^2
 

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  • #2
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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.
 
  • #3
NascentOxygen
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x*log(x)=0.1*x^2
Is log = log10? Then I'd guess about, oh, let's see, .... 10? :smile:
 
  • #4
HallsofIvy
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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/webMathematica/FunctionEvaluation.jsp?name=ProductLog. That's what I used to get the value above.)
 
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