::I understand your request, but it is a little hard to do what you are asking for, since fixing the definitions at the very beginning necessitates changes throughout the article; making changes piecemeal would lead to a lot of inconsistencies (not that the current version is consistent, when it comes to specifying whether log(z) denotes the principal value or an arbitrary logarithm or the set of all logarithms or ??? !)  Anyway, I will try to summarize the main changes.  I hope to hear what in particular you and other readers like or don't like. 

 ===Additions=== 
 *Precise definition of logarithm 
 *Precise definition of principal value of argument and of logarithm 
 *Precise definition of branch of log z 
 *Precise definition of branch cut 
 *A little more detail about differing notational conventions. 
 *Construction of branches using contour integration 
 *The precise relationship between the Riemann surface and branches 

 ===Deletions=== 
 *The entire section on "Derivation from the Taylor series"; I don't know any text that introduces log this way, but feel free to provide one if you know one. 
 *The imaginary-base logarithms.  This could be kept, but I would not recommend it. 
 *Some of the material in the conformal mapping section.  Again, some of this could be kept, though it is a little wordy for my taste. 

 ===Changes=== 
 *Using the notation Log z to distinguish the principal value from other logarithms, following Sarason's text. 
 *In general, rewriting to clarify what is being stated for the principal value, and what is being stated for arbitrary branches of log z. 
 *The section on "log z" as a multivalued function is clarified by introducing the multitude of logarithms of a number right at the beginning of the article, and by making precise the relationship between the Riemann surface and branches. 

 See my talk page for other details and rationale.