Big O Proof for Logarithmic Function

Question

I am an undergraduate student in Computer Engineering and going through one of the textbook examples, I am asked to prove that

$T(n)$ is $O(\log{}n)$

Where $T(n)= 5\log_{2} 2n +7$. I understand that this means that I must prove that $5\log_{2} 2n +7 \leq\lambda log(n)$

For some in n in a neighborhood of infinity. How exactly would I proceed with this?

Have you tried anything yourself? Why did your approaches fail? — Tom van der Zanden, Oct 14 '15 at 15:04
Looking into it, the professor and the book both describe ways of picking some arbitrary n>a and then building the inequality, eventually landing at some lambda value — pingOfDoom, Oct 14 '15 at 15:17

score 1 · Answer 1 · answered Oct 14 '15 at 15:15

1

$$5\log_2(2n)+7=5\log_2 n+5\log_2 2+7=5\log_2 n +12$$ Since for example $12\le 12\log_2 n$ for $n\ge 2$, it follows that $$5\log_2 n +12\le 5\log_2 n +12\log_2 n=17\log_2 n $$ for $n\ge 2$. This yields $O(\log n) $.

answered Oct 14 '15 at 15:15

Danny

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While this answers the question, it is common usage to not answer a question until the OP has made a serious effort at solving it on their own and (in the case of such a basic question) showed their attempts and why they failed. – Tom van der Zanden Oct 14 '15 at 15:21
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Sorry. I am sick and bored.. :) – Danny Oct 14 '15 at 15:23
@Danny Here, take these. – Raphael Oct 15 '15 at 10:18

Big O Proof for Logarithmic Function

1 Answers1