# Existence of some search algorithms

The lowest time complexity of search algorithms for sorted lists is $$O(n)=logn$$.

The lowest time complexity of sorting algorithms is $$O(n) = nlogn$$

So in order to be able to use a search algorithm for sorted lists we first must sort the list which takes min of $$nlogn$$ time complexity.Since the brute force search algorithm has a time complexity of just $$n$$ why bother sorting the list to figure out if a element is inside a list since $$n.What is the use of search algorithms of a sorted list?

In your example, sorting is worth the effort if you need to perform at least $$\Omega(\log n)$$ queries.
The effect of pre-sorting the list occurs when there are many queries. For example, when there is an unsorted list of length $$n$$, and then $$q$$ membership checks, the brute force approach takes $$O (n q)$$, whereas pre-sorting and binary search take $$O (n \log n + q \log n)$$. When $$q = 1$$, it is less efficient than brute force. But when $$q = n$$, it is $$O (n \log n)$$ versus $$O (n^2)$$, a pretty significant difference.