This answer says that in surreal numbers $\ln \omega=\omega^{1/\omega}$.

At the same time, this Wikipedia article says that transseries $\mathbb{T}^{LE}$ are isomorphic to a subfield of $No$ with its natural field structure and operations and series $x$ corresponding to the surreal number $\omega$.

Also, this article says that Hardy fields are isomorphic to $No(\omega_1)$, also a subset of surreals.

But in transseries, and in Hardy fields $\ln x\ne x^{1/x}$. How this could be the case if they are isomorphic to subfields of surreals?