### aggregate-functions

#### Complexity asymptotic relation (theta, Big O, little o, Big Omega, little omega) between functions

Let's define: Tower(1) of n is: n. Tower(2) of n is: n^n (= power(n,n)). Tower(10) of n is: n^n^n^n^n^n^n^n^n^n. And also given two functions: f(n) = [Tower(logn n) of n] = n^n^n^n^n^n^....^n (= log n times "height of tower"). g(n) = [Tower(n) of log n] = log(n)^log(n)^....^log(n) (= n times "height of tower"). Three questions: How are functions f(n)/g(n) related each other asymptotically (n-->infinity), in terms of: theta, Big O, little o, Big Omega, little omega ? Please describe exact way of solution and not only eventual result. Does base of log (i.e.: 0.5, 2, 10, log n, or n) effect the result ? If no - why ? If yes - how ? I'd like to know whether in any real (even if hypotetic) application there complexity performance looks similar to f(n) or g(n) above. Please give case description - if such exist. P.S. I tried to substitute: log n = a, therefore: n = 2^a or 10^a. And got confused of counting height of received "towers".

I won't provide you a solution, because you have to work on your homework, but maybe there are other people interested about some hints. 1) Mathematics: log(a^x) = x*log(a) this will humanize your problem 2) Mathematics: logx(y) = log2(y) / log2(x) = log10(y) / log10(x) of course: if x is constant => log2(x) and log10(x) are constants 3) recursive + stop condition

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