Logs and domains

Question

Take $y=ln(x^2)$, it's domain is all real x except 0. Now if we use the power law we end up with $y=2ln(x)$. And this functions domain is positive x. So what causes this change in domain?

Answer 1

This is because according to power law,

y= log(x^2)=2(log|x|)

So, if f(x)= 2(log(x)), log(x^2)=f|x| Hence, both are different functions and hence can have different domains.