166 CHAPTER 5 • ELEMENTARY FUNCTIONS
Our next result explains why Log (z 1 z2) = Log (z1) + Log (z2) didn't hold
for the particular numbers we chose.
As Example 5.5 and Theorem 5.2 illustrate, properties of the complex loga-
rithm don't carry over when arguments of products combine in such a way that
they drop down to -1f or rise above n. This is because of the restrictions placed
on the d omain of the function Arg. From the set of numbers associated with the
multivalued logarithm, however, we can formulate properties that look exactly
the same as those corresponding with the real logarithm.
We can construct many different branches of the multivalued logarithm func-
tion that are continuous and differentiable except at points along any preassigned
ray { reio : r > 0}. If we let a denote a real fixed number and choose the value