Binary search in sorted vector of pairs

How to apply STL binary_search to vector of pairs(key, value), given that vector is sorted by its first value(key) struct compare in the code contains two functions which compares the key(searching element) with the first element in the vector 

CPP

/* C++ code to demonstrate how Binary Search
can be applied on a vector of pairs */
#include <bits/stdc++.h>
using namespace std;

struct compare {
 bool operator()(const pair<int, int>& value,
        const int& key)
 {
  return (value.first < key);
 }
 bool operator()(const int& key,
     const pair<int, int>& value)
 {
  return (key < value.first);
 }
};

int main()
{
 // initializing the vector of pairs
 vector<pair<int, int> > vect;

 // insertion of pairs (key, value) in vector vect
 vect.push_back(make_pair(1, 20));
 vect.push_back(make_pair(3, 42));
 vect.push_back(make_pair(4, 36));
 vect.push_back(make_pair(2, 80));
 vect.push_back(make_pair(7, 50));
 vect.push_back(make_pair(9, 20));
 vect.push_back(make_pair(3, 29));

 // sorting the vector according to key
 sort(vect.begin(), vect.end());

 // printing the sorted vector
 cout << "KEY" << '\t' << "ELEMENT" << endl;
 for (pair<int, int>& x : vect)
  cout << x.first << '\t' << x.second << endl;

 // searching for the key element 3
 cout << "search for key 3 in vector" << endl;
 if (binary_search(vect.begin(), vect.end(),
        3, compare()))
  cout << "Element found";
 else
  cout << "Element not found";

 return 0;
}

Output:

KEY    ELEMENT
1    20
2    80
3    29
3    42
4    36
7    50
9    20
search for key 3 in vector
Element found

Time complexity: O(n log n) (As sort() function is used)

Auxiliary Space: O(1)