How many solutions to quadratic logarithms?

Question

For a given Log equation with a quadratic $x$, such as $$F(x)=\log(x^2)$$there appear to be two $x$ values, for every $F(x)$, a positve and a negative. However, if $F(x)$ is rearanged,$$F(x)=2\log(x)$$ then there is only one $x$ value, for every $F(x)$. How does this work?

Another way of looking at it is that the first equation, will produce two curves, one on either side of the $y$ axis, but the second will only produce one, even though they appear to be the same graph.

Answer 1

The equation $\log(a^b)=b\log a$ is valid only for positive values of $a$ (as your example clearly shows).

Answer 2

$\log(x^2)$ gives real values for non-zero $x$.

$\log(x)$ and so $2\log(x)$ gives real values for positive $x$.

So it is a question of the domain of $F(x)$

Answer 3

$\log(x^2)=2\log(\vert x\vert)$ , but not only $=2\log(x)$

So, you have two curves, one for $x<0$ and one for $x>0$