SOLUTION: Which result is correct? {{{ log( 3, ( log( 2, ( log( 2, 256 ) ) ) ) ) = x }}} {{{ log( 3, ( log( 2, ( log( 2, 2 ^ 8 ) ) ) ) ) = x }}} a) {{{ 3 ^ x = log( 2, ( log( 2, 2 ^ 8 ) )

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Question 803391: Which result is correct?


a)





b)


Where is the mistake?
Found 2 solutions by Edwin McCravy, KMST:
Answer by Edwin McCravy(20056)   (Show Source): You can put this solution on YOUR website!
Which result is correct? 



That is correct but not finished.  Here is the completion







a) 


The preceding step is wrong. Exponentiation is not
associative.  Therefore we must always use parentheses 
in such cases since  and  are
not the same. To write  would be ambiguous, and
thus it is never used.  The corrected step is



The completion is





Since the base 2 is positive and not equal to 1,
we may equate the exponents of 2:





and thus by the same rule:

x = 1


b) 



This is correct.

Edwin
Answer by KMST(5328)   (Show Source): You can put this solution on YOUR website!
I see no mistake in b), and it seems to be the most direct way to the answer.

In a), I believe that the problem is that
means that

I see those parentheses around as crucial to establish the proper meaning of the expression.

Certainly, is not the same as as you assumed to keep on going.
If , ,
while

Probably there are rules about order of operations that assign one or another meaning to , but I would forget them soon after reading them, because those expressions do not come up too often, so I will use parentheses to make my meaning clear.

In any case, I would not know how to proceed further after getting to , because I do not know of any equivalent expression.
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