Show (log⁡n)(log⁡n)=2(log⁡n)(log⁡(log⁡n))I am having difficulty understanding how this follows.(log⁡n)(log⁡n)=2(log⁡n)(log⁡(log⁡n))=nlog⁡log⁡nWhich logarithmic identities are used to go...

David Young

David Young

Answered

2022-01-21

Show (logn)(logn)=2(logn)(log(logn))
I am having difficulty understanding how this follows.
(logn)(logn)=2(logn)(log(logn))=nloglogn
Which logarithmic identities are used to go through each equality?
e.g. how do you first go from
(logn)(logn)=2(logn)(log(logn))
and then to
2(logn)(log(logn))=nloglogn
(The log base must be 2 or else this equality won't hold)

Answer & Explanation

enhebrevz

enhebrevz

Expert

2022-01-21Added 25 answers

Im
Barbara Meeker

Barbara Meeker

Expert

2022-01-22Added 38 answers

For any
1aR+,xy=aylogax,whenever LHS is defined.
Thus,
(logn)logn=2lognlog2(logn)
So your equality follows if the logarithm here is taken in base 2 and not 10 , as you wrote...
RizerMix

RizerMix

Expert

2022-01-27Added 437 answers

Let y=log2log2nn Taking the logarithm (in base 2) of both sides log2y=log2nlog2log2n Now, remember that 2log2n=n. Thus y=2log2y=2log2nlog2log2 Also recall that abc=(ab)c. Thus y=2log2nlog2log2n=(2log2n)log2log2n=nlog2log2n

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