How to solve $x = e^{1/(4x)}$?

Question

I need to find the solution to this equation :

$$ x = e^{1/(4x)} $$

Can you give me a hint ?

Answer 1

By using the Newton Method and use the function $y=x\log x-\frac{1}{4}$ $$x_{n+1}=x_n-\frac{y_n}{y'_n}$$

then $$x_{n+1}=\frac{x_n+1/4}{1+\log x_n}$$ at $x_0=1$ to get $$x1=1.25$$ $$x_2=1.226348288$$ $$x_3=1.226161263$$ $$x_4=1.226161251$$ $$x_5=1.226161251$$

Answer 2

The right answer was given above. We can also write $$x = \frac{1/4}{W(1/4)},$$

where $W$ denotes the Lambert W function defined as an inverse of $f(z) = z \exp z$, e.g. $$z = W (z\exp z) \iff f(z) = z \exp z.$$