Determine Big O of a function (Discrete math)

Question:
Determine whether the function log(n2+1) is O(logn).

Definition in paint document if needed.

Solution:

n2 + 1 < 2n2 whenever n>1

log(n2 + 1) < log( 2n2 ) = log(2) + 2log(n).

Now, log(n)>log(2) whenever n>2. Therefore

log(2) + 2log(n)<3log(n).

log(n2 + 1) ≤ 3log(n) whenever n>2.
∴Because there exists constants C and k such that log(n2 + 1)≤Clog(n) whenever n>k we can conclude log(n2 + 1) is O(logn).

Is this correct? The book says that log(n2 + 1) is not O(logn).

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Office_Shredder
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I think the book is wrong and you are right.

Dick
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Question:
Determine whether the function log(n2+1) is O(logn).

Definition in paint document if needed.

Solution:

n2 + 1 < 2n2 whenever n>1

log(n2 + 1) < log( 2n2 ) = log(2) + 2log(n).

Now, log(n)>log(2) whenever n>2. Therefore

log(2) + 2log(n)<3log(n).

log(n2 + 1) ≤ 3log(n) whenever n>2.
∴Because there exists constants C and k such that log(n2 + 1)≤Clog(n) whenever n>k we can conclude log(n2 + 1) is O(logn).

Is this correct? The book says that log(n2 + 1) is not O(logn).
I think you are right and your book is mistaken.