Is the rewritten form of ln(x2) valid?

[mindheavy]

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?

https://dl.dropbox.com/u/15809883/ln.png

 

[mfb]

ln(x^2) = 2 ln(|x|)

As sqrt(x^2)=|x|

 

[mindheavy]

Ah yes, I had forgotten about that, thank you.

 

[Mark44]

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.