(08/14/2009, 04:50 PM)jaydfox Wrote: But Bo, these zeroes are a trivial result of the "change of base" concept.

Well if you only look at the case , ; it may be trivial (though singularities not zeros) because everything stays on the real axis.

However I was considering general real bases and and the possibly non-real singularities in the region .

As you say yourself its difficult to determine whether the non-real possible singularities (which I completely specified) indeed exist or cancel out by appropriate choices of the logarithm. I showed that indeed has complex singularities that dont cancel out. And also that there are complex singularities for other bases that dont cancel out.

The example of shows also that your suggested path-continuation does not work. If you have branch points then different paths to the same point may result in different values of . To keep it continuous you have to specify cuts.