@arjun sir it is possible with merging algorithm also.This is given in cormen second edition page number 38 as given below:

b) Show that sublists can be merged in theta(nlog(n/k)) worst case time.

but only difference is here n/k sorted sublist and each of lenth k but in our question logn sorted list and each of length n/logn and total number of elements in both the cases is n only

so we can say that time complexity=total number of elements * log (number of sublist)

=n * log(n/k)

that is why for this question n * log(logn)

(it is also true in simple case when we have array of n elemets then merge sort divide the array in such a whether until all array elememts split in to single single elements and then using merging algorithm we started merging by taking 2 elements at a time because single element is already sorted and here time complexity =theta(nlogn) that means time complexity =total number of elements * log (number of sublist)

here number of sublist is n because array is divided in such a whether until each list contain 1 element and then apply merging.)

therefore in this question we can also consider that our array is divided in such a whether until each list contain n/logn elements because these lists are sorted given in question then apply merging by taking 2 sorted list then we will get time complexity=

total number of elements * log (number of sublist)

=n * log(logn)