Given the existence and uniqueness theory:
If $f$ is Lipschitz continuous over some region $D$, then there is a unique solution to the initial value problem (IVP):
$u'(t) = f(u,t), \hspace{5mm} u(0)=\eta$
at least up to time $T^*=min(t_1,t_0+a/S)$ where,
$ S = \max\limits_{(u,t)\in D} |{f(u,t)}| $.
What I'd like to do (riffing off a previous question I asked, here), is to use the E&U theorem to show that:
$u'(t) = log(u(t))$
$u(0) = 2$
Actually has a unique solution for any $t \geq 0$.
My strategy would be to show that $f$ is Lipschitz continuous inside a domain that the solution never leaves, but I'm not quite sure how.