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Really Simple Logarithmic Question

  1. Sep 11, 2011 #1
    1. The problem statement, all variables and given/known data

    1. f(n) = log(n) ^ (log(n)). Simplify for Big-O notation.

    3. The attempt at a solution
    1.

    I'm just trying to prove a big-O notation for one of my courses, and this was simplified to

    f(n) = n^log(log(n))

    And I'm having a hard time seeing why. Any guidance would be appreciated.
     
    Last edited: Sep 11, 2011
  2. jcsd
  3. Sep 11, 2011 #2

    Stephen Tashi

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    Science Advisor

    [tex] \log{f(n)} = \log( (\log n)^{\log{n} } ) [/tex]
    Using [itex] \log(a^b) = b\log(a) [/itex]
    [tex] \log f(n) = \log(n) \log( \log(n)) [/tex]

    Let's say the logs are base 10. Then
    [tex] 10^{\log f(n)} = 10^{\log(n) \log(\log(n)} [/tex]
    [tex] f(n) = (10^{log(n)})^{\log(\log(n))} [/tex]
    [tex] f(n) = n^{log(log(n))} [/tex]
     
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