Really Simple Logarithmic Question

1. Sep 11, 2011

l46kok

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. Sep 11, 2011

Stephen Tashi

$$\log{f(n)} = \log( (\log n)^{\log{n} } )$$
Using $\log(a^b) = b\log(a)$
$$\log f(n) = \log(n) \log( \log(n))$$

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