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Log[x] is not singlevalued function in complex domain.
It is important which branch is chosen.
For example, Mathematica gives:
In[11]:=
Log[E^(1-I)]
Out[11]=
1-I

But,

In[10]:=
Log[E^(1-5I)]
Out[10]=
(1-5 I)+2 I \[Pi]
so  Log[E^x] is not x.


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