When does derivative of complex Log exist?

[mnb96]

Hello,
if I have a function $$f:\mathbb{R}\rightarrow \mathbb{C}$$, what are the conditions on f for which the derivative of the Logarithm exist?

$$\frac{d}{dx}\mathrm{Log} (f(x))$$ exists?

Note that here I defined:

$$Log(z)=\log |z| + Arg(z)$$, where $-\pi< Arg(z) \leq \pi$

 

[Cyosis]

Try the Cauchy-Riemann equations.

 

[mnb96]

Ok...so, given the fact that Log(z), (with z complex) as defined above, is differentiable in the region:

$$|z|>0$$
$$Arg(z)\in (-\pi,\pi)$$

In order for Log(f(x)) to be differentiable, I need to have that;

1) the image of f(x) must be in that region (the one where Log is differentiable)
2) f must satisfy the C-R equations

Is this correct?