Question 1198695
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Aplique a propriedade do logaritmo 

Log2 (4x2)
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        This problem, harmless at first sight,  has a huge underwater stone,  like a trap,

        which was overlooked by other tutors.



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We start from this expression  {{{log(2,(4x^2))}}}.


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

    {{{log(2,(4x^2))}}} = {{{log(2,(4))}}} + {{{log(2,(x^2))}}} = {{{2 + log(2,(x^2))}}}.



Now,  {{{log(2, (x^2))}}} = {{{2*log(2,(abs(x)))}}}  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,  {{{log(2,(4x^2))}}} = {{{2 + 2*log(2,(abs(x)))}}}.


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,  {{{abs(x)}}},  under the logarithm 
in the final expression.


Then (and only then) the identity 

    {{{log(2,(4x^2))}}} = {{{2 + 2*log(2,(abs(x)))}}}.


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

Solved.


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The meaning of this assignment is to simplify the given expression accurately over the entire domain.