Çözüldü Principal Argument and Logarithm of Complex Numbers - Euler Formula

A test adaptation of a solved problem from the University of California - Berkeley:

What is the principal argument of (1 + i)^i ?
A) log(2)
B) log(i)
C) π / 4
D) log(2) / 2
E) π

1 + i = z
1·1 + 1·i = z
[ (1^2 + 1^2)^0.5 ]·{ e^[ i·arctan(1 / 1) ] } = z
(√2)·{ e^[ i·arctan(1) ] } = z
(√2)·[ e^(i·π / 4) ] = z = (1 + i)
{ (√2)·[ e^(i·π / 4) ] }^i = z^i = (1 + i)^i
[ (√2)^i ]·[ e^(-π / 4) ] = (1 + i)^i
[ 2^(i / 2) ]·[ e^(-π / 4) ] = (1 + i)^i....(I)
2^(i / 2) = e^t
(i / 2)·log(2) = t....(II)
log(2) = 2t / i ⇒ 2 = e^(2t / i) = e^[ (2 / i)·t ]....(III)
Substituting (II) into the right hand side of (III);
2 = e^[ (2 / i)·(i / 2)·log(2) ]
2 = e^log(2)
2^(i / 2) = e^[ (i / 2)·log(2) ] = cos[ log(2) / 2 ] + i·sin[ log(2) / 2 ]
Arg[ (1 + i)^i ] = log(2) / 2
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Original Problem:

https://i.ibb.co/5R4mc2d/Berkeley-Complex-Number.png
https://math.berkeley.edu/~giventh/hw4.pdf
(Page 2, Question IV.16.1.)