static variable link error

You declared static string theString;, but haven't defined it.

Include

string Log::theString;

to your cpp file

You must define the statics in the cpp file.

Log.cpp

#include "Log.h"
#include <ostream>

string Log::theString;  // <---- define static here

void Log::method(string arg){
    theString = "hola";
    cout   << theString << endl; 
}

You should also remove using namespace std; from the header. Get into the habit while you still can. This will pollute the global namespace with std wherever you include the header.