Can anyone help me understand this equation?

$\log (n) = O(n^c)$ (for any $c>0$)

Does it mean that $O(\log (n)) < O(n^c)$ (for any $c>0$)?


Please also prove that $\log (n) = O(n^c)$ is true.

  • 2
    $\begingroup$ You just have to show that logarithms grow slower than polynomials. $\endgroup$ – Pål GD Apr 20 '15 at 8:28
  • $\begingroup$ Attend the definition. $\endgroup$ – Raphael Apr 20 '15 at 8:51
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    $\begingroup$ No, it means that, for any $c>0$, there's a constant $k$ such that, for large enough $n$, $\log n < kn^c$. Just like any other statement of the form $f(n)=O(g(n))$. $\endgroup$ – David Richerby Apr 20 '15 at 10:46

It's not actually an equation $f = O(g)$ is a lazy shorthand that should be written $f \in O(g)$. So if you look back at the definition of $O$, you should be able to see what $\log n \in O(n^{c})$ for any $c > 0$ means:

For every $c > 0$, there exists $n_{0} \geq 0$ and $k \geq 0$ such that $\log n \leq k\cdot n^{c}$ for all $n \geq n_{0}$.

Always remember that $O(\cdot)$ describes a set, $O(\log n) < O(n^{c})$ doesn't actually make sense (unless you make up a special meaning for $<$, but then no-one will know what you're talking about). You could say $O(\log n) = O(n^{c})$, as equality for sets has a understood meaning, though of course this statement in particular would be false.

As John Kugelman points out in the comments below, normal set relations do make sense, so $O(f) \subset O(g)$, $O(f) \subseteq O(g)$, etc., make sense.

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    $\begingroup$ $O(\log n) \subset O(n^c)$ is sensible. $\endgroup$ – John Kugelman Apr 20 '15 at 14:31
  • $\begingroup$ Could you add a prove for log(n)=O(n^c), please? $\endgroup$ – Xin Apr 21 '15 at 4:57
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    $\begingroup$ @Xin the two links in the comments on the question have all the tools you need to work out the proof, have a go, see where you get stuck, walk away from it for a day, try again, then come back and you'll get a lot more out of a more precise question at that point, than just getting someone else's version of the proof now. $\endgroup$ – Luke Mathieson Apr 21 '15 at 12:46
  • $\begingroup$ @Xin Hint: you won't motivate people to help you if you demonstrate that clearly that you are not interested in understanding but only in having something to hand in tomorrow. $\endgroup$ – Raphael Apr 21 '15 at 13:00

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