Ambiguous conditions for $x$ in logarithm? I have just realised that there is a little ambiguity in defining conditions for $x$ in logarithm. Let me illustrate it on a simple example:

$\log{x}$ is valid for $x&gt;0$ ,

$2\log{x}$ is also valid only for $x&gt;0$ ,

but $\log{x^2}$ is valid for both $x&gt;0$ and $x&lt;0$ .

How is that possible when $2\log{x} = \log{x^2}$ ?
Note that logarithm identities are true only when both sides of the equation are defined, so $2\log x = \log x^2$ for $x &gt; 0$, and not when $x &lt; 0$.

This thinking could lead to things like $-1 = \sqrt{-1} \cdot \sqrt{-1} = \sqrt{-1 \cdot -1} = \sqrt{1} = 1$, which is obviously false. The problem here is that $\sqrt{ab}=\sqrt{a}\cdot\sqrt{b}$ only when $a,b&gt;0$.
As well known language do not follow the rules of logics. "If you behave well 
you will have an ice cream" according to logic does not imply that if you behave badly you will not have the ice cream, but any child knows that in language this is precisely what it means. One could argue that language reflect the way we reason, logics the way we should reason. No students, even the ones in math would not understand what you mean when you say $\log(x)$ is defined for positive $x$ and then $\log x^2 =2 \log x$.