In this paper, we study the (possible) solutions of the equation exp * (f ) = g, where g is a slice regular never vanishing function on a circular domain of the quaternions H and exp * is the natural generalization of the usual exponential to the algebra of slice regular functions. Any function f which satisfies exp * (f ) = g is called a * -logarithm of g. We provide necessary and sufficient conditions, expressed in terms of the zero set of the "vector" part gv of g, for the existence of a * -logarithm of g, under a natural topological condition on the domain Ω. By the way, we prove an existence result if gv has no non-real isolated zeroes; we are also able to give a comprehensive approach to deal with more general cases. We are thus able to obtain an existence result when the non-real isolated zeroes of gv are finite, the domain is either the unit ball, or H, or D and a further condition on the "real part" g 0 of g is satisfied (see Theorem 6.19 for a precise statement). We also find some unexpected uniqueness results, again related to the zero set of gv, in sharp contrast with the complex case. A number of examples are given throughout the paper in order to show the sharpness of the required conditions.