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Big-O With Change of Base for Log

  1. Jul 12, 2009 #1
    1. The problem statement, all variables and given/known data
    Hi, I am trying to prove for a>1, b>1 that f(x) = O(log_b(x)) then f(x) = O(log_a(x)).
    [Hint: log_b(x) = log_a(x)/log_a(b)).

    2. Relevant equations

    3. The attempt at a solution
    Assuming |f(x)| <= C|log_b(x)| true for x>k,
    f(x) <= log_a(x)/log_a(b)| for all x>1, C=1
    log_a(b)*f(x) <= log_a(x) all x>1 C= 1/log_a(b)
    f(x) <= log_a(x) x>1 C=1/log_a(b)
    f(x) is O(log_a(x))

    Now here comes my problems with most big-O problems, since the right side is being divided by something I should just multiply it out to other side:

    log_a(b)*f(x) <= log_a(x) I know that log_a(b) on left side would not matter except that it might be
    C= 1/log_a(b) and k =1

    According to my textbook/notes, C = 1/log_a(b), I still do not see why though, because if I multiply both sides by log_a(b) why C is like dividing both sides log_a(b), if someone could explain it, I would appreciate it thank you very much.

    Please Let me Know if this correct, thank you very much.

  2. jcsd
  3. Jul 13, 2009 #2

    Sorry, I don't follow that list of inequalities at all.

    Can you try it more clearly, perhaps like this.

    Given: There exist C and k such that |f(x)| < C |log_b (x)| for all x > k.

    Prove: There exist C' and k' such that |f(x)| < C' |log_a (x)| for all x > k'.

    Your k' can be the same as k. Your C' will be expressed in terms of C and log_a (b).
  4. Jul 13, 2009 #3

    Prove : a>1, b>1
    f(x) = O(log_b (x)) then f(x) = O(log_a (x))

    |f(x)| < C |log_b (x) | for all x>k
    |f(x)| < |log_a (x) / log_a (b) | for all x>1

    then f(x) = O(log_b (x)) = O(log_a (x)) when k=1 C=1/log_a (b)
    I am just hoping to know if thats correct.

    Thank You Very Much.
  5. Jul 13, 2009 #4
    I don't know how you are obtaining the last line quoted. It is invalid reasoning.


    Let a=10, b=e. Use ln x to denote log base e of x, and let log x denote log base 10 of x.

    Let f(x) = 1 + 4 ln x.

    Then f is O(ln x), because f(x) < 5 ln x when x > 3. (C=5, k=3.) (Verify with a graph if you want.)

    Your next step says f(x) < log x / log e for x > 1. There is no reason for this to be true, and in fact it is false. (Verify with a graph if you want.)
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