Compound interest ($r$ in terms of $\frac{dA}{dt}$)
You are not doing anything wrong. Since
$$
\lim_{r \to 0} \frac{\ln(1+r)}{r} = 1
$$
(to see why: use l'Hôpital's rule or Taylor approximation) holds, for small numbers $r$ 
$$
\ln(1+r) \approx r.
$$