Hello @Gianluca Gorni , my software gives me a different output:

In[24]:= D[Log[RealAbs[x]], x] // Simplify

Out[24]= Derivative[1][RealAbs][x]/RealAbs[x]

Do you know why? I'm using Mathematica 10.2.

Hello @Carlo Barbieri , is there a way to tell to Mathematica that x is real? If I do this, Mathematica'll give me Log[Abs[x]] (I hope).

I'm not sure what you ask for is the correct result:

In[86]:= Integrate[1/x, {x, 1, z}]
Out[86]= ConditionalExpression[Log[z], Re[z] > 0 || z \[NotElement] Reals]

Remember that in general Mathematica treats symbols that you haven't otherwise specified as complex. So if you replace z with I, you will find I Pi/2 which is Log[I]. So in general the result of that integral for negative real z doesn't exist, but it does exist for arbitrary complex numbers, so you just can't take the Abs there.

This is where things start to get interesting: Exp[I Pi/2 + 2 k Pi] == I for all k, so actually you can't really say that Log is the inverse of Exp in the complex plane. This will get you into the amazing territory of Riemann surfaces, but I don't want to go any further. Maybe you can read this wikipedia article.