We start from this expression  .


In this expression, x can be any non-zero number, negative or positive.


In other words, the domain, where the expression is defined / (makes sense), 
is the set of all real non-zero numbers {R \ {0} }.


In this domain

     =  +  = .



Now,   =   is valid for all values of x in the domain, positive or negative,
excluding the zero value of x.  Notice the absolute value sign under the logarithm.


Therefore, in the entire domain,   = .


It does not matter if you take the factor of "2" outside the parentheses or not.


What is REALLY IMPORTANT, is to use the sign of absolute value,  ,  under the logarithm 
in the final expression.


Then (and only then) the identity 

     = .


is valid on the entire domain, which is  {R \ {0} }, the set of all real non-zero numbers.