A Note on Schanuel's Conjectures for Exponential Logarithmic Power   Series Fields
  In [1], J. Ax proved a transcendency theorem for certain differential fields of characteristic zero: the differential counterpart of the still open Schanuel's conjecture about the exponential function over the field of complex numbers [11, page 30]. In this article, we derive from Ax's theorem transcendency results in the context of differential valued exponential fields. In particular, we obtain results for exponential Hardy fields, Logarithmic-Exponential power series fields and Exponential-Logarithmic power series fields. 
Salma Kuhlmann, Mickael Matusinski, Ahuva C. Shkop
math.LO math.AC