I am working with my homework, where I am asked to solve an equation and provide the answer analytically. $$e^{1-3x}-x=0$$ I can get an numerical solution ($x\approx0.54$) but I am really struggling with the analytical solution. I am starting to believe that this equation can't be solved analytically but in am not sure. If the Lambert function can be applied, how can I manipulate the equation in order to get an answer?
In advance thanks!
$$e^{1-3x}-x=0$$ $$e^{1-3x}=x$$ $$1=xe^{3x-1}$$ $$3e=3xe^{3x}$$ Since the RHS is now in the form $ze^z$, we can apply the Lambert W: $$W(3e)=3x$$ $$x=\frac{W(3e)}3=0.539214\dots$$