How to use omega function

Question

In my work I have faced an optimization problem to resolve which I have to solve the following expression (not the one that I actually work with, but structurally they are the same): $\ln(x) + x = 10$. I know that to get $x$ I can use the Omega function aka Lambert W Function, however, I can't understand how it works. Thus, I ask for help of those of you who know how to solve this kind of equations (showing as much steps as you can) in order to give me reference point for further studying of this topic.

Answer 1

The $W$-function inverts $z = we^w$, $w \ge -1$. You need to get a product of the form $we^w$ in your expression. Use the laws of exponents to find $$e^{\ln x + x} = xe^x$$ so that $xe^x = e^{10}$. This leads to $x = W(e^{10})$.

Answer 2

From your initial equation:

$$\ln{x} + x = 10$$

you can exponentiate both sides:

$$e^{\ln{x} + x} = e^{10}$$

which is the same as:

$$e^{\ln{x}} \cdot e^x = e^{10}$$

which is the same as (because $e^x$ and $\ln{x}$ are inverses):

$$x \cdot e^x = e^{10}$$

From there, you can use the definition of the Lambert W function:

$$x = W(e^{10}) \approx 7.92$$