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I have stumbled into this equation $0.1=4.8626\cdot xe^{-4.472\cdot x}$

I tried to take the natural logarithm for both side but it didn't help as it will result in $\ln x+x$ which I can't solve. Can someone please show me how to solve this equation in steps (it doesn't have to be the same constants as in my question ) I tried to search for a method to solve it but apparently it's related to lambert w function and the method to solve it is not really clear to me.

Thomas Andrews
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Jacob
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2 Answers2

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This requires the Lambert W-function. Change the equation to:

$$0.1\cdot \frac{-4.472}{4.8626} = (-4.472x)e^{-4.472x}$$

Then:

$$x=\frac{1}{-4.472}W\left(0.1\cdot \frac{-4.472}{4.8626}\right)$$

There are two possible values for this $W$ argument.

Wolfram alpha gave me one solutions as $x=0.0227685\dots$ but I could not figure out how to get the other value or $x.$

Gary's value, in comments, $x=0.8257319539\dots,$ seems to work, too, but I don't know how he got it.

Thomas Andrews
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    The other value is $x=0.8257319539\ldots$. – Gary Feb 27 '20 at 17:35
  • Thanks. How did you get that, @Gary – Thomas Andrews Feb 27 '20 at 17:47
  • The equation has two possible values and, Gray's answer is the one that I'm looking for, but how would I obtain that value with the hand calculations because I've used Symbolab and Matway and I as able to obtain the two values but not the steps – Jacob Feb 27 '20 at 17:48
  • You'll have to look at the Wikipedia pages for the W function to figure out how to calculate it. If $-1/e<z<0$ then there are two values of $w=W(z),$ one with $-1<w<0$ and the other with $w<-1$ @Jacob – Thomas Andrews Feb 27 '20 at 17:53
  • @ThomasAndrews how did you solve the W(0.1*-4.472/4.8626) – Jacob Feb 27 '20 at 17:53
  • I used Wolfram alpha, specifically: LambertW(-0.091963477831702) @Jacob – Thomas Andrews Feb 27 '20 at 17:54
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    Oh, also in Wolfram Alpha, you get the $W_{-1}$ branch, and hence the other answer, by computing ProductLog(-1,-0.091963477831702) – Thomas Andrews Feb 27 '20 at 18:01
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Update: I've came into a method to find the two solutions of this equation using any regular scientific calculator for that(I used Casio FX991es calculator) you need two write the equation as it is and use [Shift+solve] and let the calculator find the first solution then do the same process again but when hitting [shift+ solve] give the calculator an initial value of x that is larger than the other one and keep trying till it hits the second solution. That way you can find the two solutions of the equation without using any online calculator or use lambert's method.

Jacob
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