TITLE: Why are the domains for $\ln x^2$ and $2\ln x$ different?
QUESTION [10 upvotes]: If I have a function like this $f(x)=2 \ln(x)$ and I want to find the domain, I put $x>0$. But if I use the properties of logarithmic functions, I can write that function like $f(x)=\ln(x^2)$ and so the domain is all $\mathbb{R}$ and the graphic of function is different. Where is the mistake?

REPLY [4 votes]: First, you could use $\ln x$ to define functions with different domains as long as $\ln x$ is defined in that domain.
Second, the rule  $\ln x^n=n\cdot \ln x$ is a bit sloppy. It should always be pointed out that $x>0$. Likewise, $\ln ab=\ln a+\ln b$, only if $a,b>0$.