induction e1.
intros.
simpl.
specialize (expnat_2_nz e2). intros.
omega.
induction e2;
intros.
omega.
simpl.
specialize (IHe1 e2).
omega.
Qed.

Theorem lognat_correct_eq : forall(b : nat)(pf : (expnat 2 b) <> 0),
(lognat pf) = b.