Evaluate the complex logarithmic function $\log(1-i)$ Evaluate $\log(1-i)$.

I have $\log(1-i) = \operatorname{Log}|1-i|+i\operatorname{Arg}(1-i)+i2k\pi$

How do I compute $\operatorname{Log}|1-i|$?
First, find $|1-i|$. That's positive real ($\left|a+bi\right|=\sqrt{a^2+b^2}$). Then find its log.
Firstly, you compute the modulus.  Second, determine the argument.     The format is $\log(r)+i
arg(2\pi n)$. 
This is how complex logarithm is determined.