Prove (log n)^2<=2^n by induction

zaviknuogg 2022-09-24 Answered
Prove ( log n ) 2 2 n by induction
I've trying to solve this for quite a while now, but not being able to finish the proof.
Prove using induction that ( log n ) 2 2 n
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Answers (2)

Lilliana Mason
Answered 2022-09-25 Author has 11 answers
For n = 1 , 2 , 3 it's clearly true.
Assume that it holds for n = k: log k 2 n / 2
Then
log ( k + 1 ) = log k log ( k + 1 ) log k = log k ( 1 + log ( 1 + 1 k ) log k ) log k ( 1 + 1 k log k ) 2 n / 2 2 1 / 2 = 2 ( n + 1 ) / 2 ,
as
1 + 1 k log k 2 , for  n 3
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videosfapaturqz
Answered 2022-09-26 Author has 3 answers
Try this roadmap:
log n < n for all n
n 2 2 n for n 3
handle the case n = 3 separately.
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