I'm interested in learning more about continuous-time modeling in economics. I've been studying a recent paper in economics that models a simple control problem that yields the following equation:
$$C_t = \bar C \exp \left( - \frac{1}{\gamma} \int_t^\infty (r_s - \rho) ds \right)$$
I understand how this equation is derived. In the paper, the authors claim (without any derivations) that is is equivalent to:
$$ d \log C_t = - \frac{1}{\gamma} \int_t^\infty d r_s ds $$
I'm a novice with continuous-time models and I'm at a loss as to how this derivation is completed. More than that, I'm not even sure what "type" of operation I should be trying to learn. This looks to me like a total differential combined with Liebniz rule, but I would like a solid reference.
HINT
Note that $$ d \ln C_t = \frac{C_t'}{C_t} $$ and $$ C_t' = C_t \frac{d}{dt} \left[- \frac{1}{\gamma} \int_t^\infty (r_s - \rho) ds\right] = -\frac{1}{\gamma} \frac{d}{dt} \left[\int_t^\infty (r_s - \rho) ds\right] $$