Question

Bring the following functions in the increasing order according to their growth rate: ? 22 ? , log(? 2 ), ? log(?) , √?, (log ?) 2 , 2 log? , 2 2? , ?2 √? , ?!

All logarithms are to the base 2

Answer 1

We have to arrange following in increasing order of growth rate

n , 22n, log(n2), n log(n) , √? , (log n)2 , 2 log n, 22n , n2√? , ?!

now to check for increasing order of growth rate we can apply a simple method . We will put very large values of n and then check for which f(n) is higher then we will arrange accordingly. You may argue that may be that value of n<n0 . But it is not a common thing. generally for these type of functions having no values it is so that value of n0 is not very much.

so let n=101000

now we will put this very large value for each function given here.

1)  n=101000

2) 22n = 22 * 101000

3) log(n2) = 2*log(n) = 2 *log (101000) = 2*1000 =2000

4) n log(n) = 101000 log (101000) = 101000*1000 = 101003

5) √? =  (101000)1/2 = 10500

6) (log n)2 = (log 101000)2 = (1000)2 = 106

7) 2 log n = 2 * log(101000) = 2 * 1000 = 2*103

8) 22n =

9) n2√? = (101000 * 2 ) 10500 = 102500

10 )n ! = (101000) !

Therefore based on above observations we can arrange functions from lower growth rates to higher rates

2 log n < log(n2) < (log n)2 < √? < n = 22n< n log(n)< n2√? < n !< 22n