Theorem 5.3.  Let be nonzero complex numbers. The multivalued function obeys the familiar properties of logarithms:

(5-17)            ,  

(5-18)            , and

(5-19)            .  

Proof of Theorem 5.3.

Identity (5-17) is easy to establish:  Using Identity  (1-38) in Section 1.4 concerning the argument of a product (and keeping in mind we are dealing with sets of numbers), we write  

               

Identities (5-18) and (5-19) are left as exercises.

(c) 2006 John H. Mathews, Russell W. Howell