$x+\ln(x)=0$, what is $x$?

Question

My friend came across this strange equation and I cant find mathematical way to find $x$ without drawing $x$ and $-\ln(x)$ and see that they come across at almost $x=0.5$.

Can any one help?

Answer 1

$$x+\ln x=0$$ $$e^{x+\ln x}=e^0=1$$ $$xe^x=1$$

This can be solved by lambert $W$:

$$x=W(1)$$

There is a special name to this constant, it is called the omega constant.

You can find the numerical approximation via Newtons method.

Let $x_1=0.5$

And

$$x_{n+1}=x_{n}-\frac{x_n+\ln x_n}{1+\frac{1}{x_n}}$$

As

$$n \to \infty$$

Then you get closer to the approximate value:

$$x=.56714329...$$