the natural log function

I'm reading back over a calculus book getting ready for an exam and I'm seeing a note that I don't understand.

It says to make sure, when rewriting a ln function that the domain is the same, then it provides an example of when it's not the same, yet says nothing more. Is this rewritten form valid?

 PhysOrg.com science news on PhysOrg.com >> Front-row seats to climate change>> Attacking MRSA with metals from antibacterial clays>> New formula invented for microscope viewing, substitutes for federally controlled drug
 Recognitions: Homework Help Science Advisor ln(x^2) = 2 ln(|x|) As sqrt(x^2)=|x|

Mentor

the natural log function

The difference in the graphs is entirely due to the domains of the two different functions.

ln(x2) is defined for all real x ≠ 0.
2 ln(x) is defined only for x > 0.

The rules for logarithms contain limitations on the values of the arguments. For example, ln(a*b) = ln(a) + ln(b), where a > 0 and b > 0. Note that it is possible for ln(a * b) to be defined even though the right side is undefined. This can happen when both a and b are negative.