A new scheme for deriving compact expressions of the logarithm of the exponential product is proposed and it is applied to several exponential product formulas. A generalization of the Dynkin–Specht–Wever (DSW) theorem on free Lie elements is given, and it is used to study the relation between the traditional method (based on the DSW theorem) and the present new scheme. The concept of the operator functional derivative is also proposed, and it is applied to ordered exponentials, such as time-evolution operators for time-dependent Hamiltonians.