Missing root

Assuming I have the following equation:

x^2 = 9

this can easily be solved by taking the square root, to give +/- 3

Suppose instead I choose to use logs.

ln (x^2) = ln 9

2 ln x = ln 9

ln x = 0.5 ln 9

ln x = ln (9^0.5)

We know we cannot take logs of negative numbers, so all we get is:

ln x = ln 3

so x = 3

How can we get the negative root using this method?

...

Since $\ln x$ is only defined for $x>0$, $\ln x^2 = 2\ln x$ should be rewritten as

$\ln x^2 = 2\ln |x|$

so that x can be both positive and negative.

Does that mean whenever you see $\ln (x)$, there's always going to be two possible answers?