Define .
.
Working on the RHS:
, where .
Working on the LHS:
.
So .
Clearly and .
Therefore, .
Alternatively, just use the fact that .
If
Then
where .
To start with, do the same thing you would if it were a "real numbers" problem. Add 2 to both sides to get
and then
Now, if a complex number, z, is written in "polar form", then
So just write 2+ 2i in "polar form". Don't forget that whatever is, adding to it doesn't change z but will give different logarithms. That the reason for the "all solutions".