How to solve $0.1=4.8626\cdot xe^{-4.472\cdot x}$

Question

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.

Answer 1

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.

Answer 2

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.