Number theory problem- 27 Nov'15

Question

Let !!a = log_3 log_3 2!!. An integer !!k!! satisfying !!1 < 2^((-k + 3^(-a))) < 2!! must be less than?

!!a)" "0!!

!!b)" "1!!

!!c)" "2!!

!!d)" "!!None

Solution

In the given term, !!a = log_3 log_3 2!!

!!=> 3^a = log_3 2!!

!!=> 3^(-a) = log_2 3 = 1.58!!....... (i)

Now,

!!1 < 2^((-k + 3^(-a))) < 2!!

Taking !!log!! with base !!2!! both sides, we get

!!=> log1 < (-k + 3^(-a))log_2 2 < log_2 2!!

!!=> 0 < -k + log_2 3 < 1!! ... using equation (i)

!!=> k < log_2 3!!

Hence, !!k < 2!!

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