Can I take log on both sides of inequality such way? f ( n ) ≤ c n k and get log ⁡ ( f ( n ) ) ≤ k c log ⁡ nI know that by rules the result inequality should be like this log ⁡ ( f ( n ) ) ≤ log ⁡ ( c n k )but I read in book where inequality (1) → in inequality (2) and want to know which way it was doneEdit: this is not general case, the questions relates to the algorithms and O-notation

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Can I take log on both sides of inequality such way?  f  (  n  )  ≤  c      n    k  and get  log  ⁡  (  f  (  n  )  )  ≤  k  c  log  ⁡  nI know that by rules the result inequality should be like this  log  ⁡  (  f  (  n  )  )  ≤  log  ⁡  (  c      n    k    )but I read in book where inequality (1)   → in inequality (2) and want to know which way it was doneEdit: this is not general case, the questions relates to the algorithms and O-notation

upornompe · Accepted Answer

You can take the logarithm on both sides of the inequality, if you know the numbers are positive. This produces   log  ⁡  (  f  (  n  )  )  ≤  log  ⁡  (  c      n    k    )However   log  ⁡  (  c      n    k    ) is not the same thing as   k  c  log  ⁡  n, so your second inequality doesn&#039;t follow. What you do have is   log  ⁡  (  c      n    k    )  =  log  ⁡  (  c  )  +  k  log  ⁡  (  n  ), so you can get  log  ⁡  (  f  (  n  )  )  ≤  log  ⁡  (  c  )  +  k  log  ⁡  (  n  )

Yahir Crane · Accepted Answer

yes you can since   l  o  g is an increasing function and hence preserves the inequality. the main condition is that both sides lies in the domain of   l  o  g i.e. they are positive  log  ⁡  (  f  (  n  )  )  ≤  log  ⁡  [  c      n    k    ]  =  log  ⁡  c  +  log  ⁡      n    k    =  log  ⁡  c  +  k  log  ⁡  nIf   c  &amp;gt;  1 then   log  ⁡  c  &amp;gt;  0 and so  log  ⁡  (  f  (  n  )  )  ≤  k  log  ⁡  n  ≤  k  c  log  ⁡  n

Can I take log on both sides of inequality such way? f ( n ) ≤

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