How to solve $x\ln(x) = 1$ with the Lambert W function or Newton's method

Question

I want to find the value of $x$ in the below equation using the Lambert W function or Newton's method.

$$x\ln(x) = 1$$

I hope give me a details of the solution.

Thank you.

Answer 1

To solve it using the Lambert W function is straightforward.

Hint: Use the substitution $x=e^u$.

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To solve it using Newton's Method, you have to apply: $$x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)} \tag{1}$$ Using: $$f(x)=x\ln{x}-1 \implies f'(x)=\ln{x}+1$$ Substituting into $(1)$, you obtain: $$x_{n+1}=x_n-\frac{x_n\ln(x_n)-1}{\ln(x_n)+1} \tag{2}$$ It just remains to choose a suitable value for $x_0$ and apply $(2)$ recursively.