  Let $\mathbb{T}$ be the differential field of logarithmic-exponential
transseries. We show that the expansion of $\mathbb{T}$ by its natural
exponential function is model complete and locally o-minimal. We give an
axiomatization of the theory of this expansion that is effective relative to
the theory of the real exponential field. We adapt our results to show that the
expansion of $\mathbb{T}$ by this exponential function and by its natural
restricted sine and restricted cosine functions is also model complete and
locally o-minimal.
