The $\ast$-exponential as a covering map

We employ tools from complex analysis to construct the $\ast$-logarithm of a quaternionic slice regular function. Our approach enables us to achieve three main objectives: we compute the monodromy associated with the $\ast$-exponential; we establish sufficient conditions for the $\ast$-product of two $\ast$-exponentials to also be a $\ast$-exponential; we calculate the slice derivative of the $\ast$-exponential of a regular function.