Deriving definition of the complex logarithm Given that: 

$$z = Re^{i\theta} = R(a + bi) = R\left( \cos(\theta) + i\sin(\theta) \right)$$ 

In its polar form. 

$$\log(z) = \log(R) + i\theta$$

$$|z| = \sqrt{(Ra)^2 + (Rb)^2} = R\sqrt{a^2 + b^2} = R\sqrt{\cos^2(\theta) + \sin^2(\theta)} = R$$

$$\log(z) = \log(|z|) + i\theta$$ 

But is taking the absolute value justified?
how do you mean justified? as in is it correct? in either case lets look at
$$
z = R\mathrm{e}^{i\theta}\implies z^* = R\mathrm{e}^{-i\theta}
$$
thus 
$$
|z| = \sqrt{zz^*} = \sqrt{ R\mathrm{e}^{i\theta}R\mathrm{e}^{-i\theta}} = \sqrt{R^2} = R
$$
thus
$$
z = |z|\mathrm{e}^{i\theta}
$$
which you take logs and obtain the result you desire..but I do not know if this enlightens any further.
how do you mean justified? as in is it correct? in either case lets look at
$$
z = R\mathrm{e}^{i\theta}\implies z^* = R\mathrm{e}^{-i\theta}
$$
thus 
$$
|z| = \sqrt{zz^*} = \sqrt{ R\mathrm{e}^{i\theta}R\mathrm{e}^{-i\theta}} = \sqrt{R^2} = R
$$
thus
$$
z = |z|\mathrm{e}^{i\theta}
$$
which you take logs and obtain the result you desire..but I do not know if this enlightens any further.