The efficiency of an algorithm

The efficiency of an algorithm

I am having a hard time understanding the efficiency of an algorithm and
how do you really determine that, that particular sentence or part is lg
n, O (N) or log base 2 (n)?
I have two examples over here.
doIt() can be expressed as O(n)=n^2.
First example.
i=1
loop (i<n)
doIt(…)
i=i × 2
end loop
The cost of the above is as follows:
i=1 ... 1
loop (i<n) ... lg n
doIt(…) ... n^2 lg n
i=i × 2 ... lg n
end loop
Second example:
static int myMethod(int n){
int i = 1;
for(int i = 1; i <= n; i = i * 2)
doIt();
return 1;
}
The cost of the above is as follows:
static int myMethod(int n){ ... 1
int i = 1; ... 1
for(int i = 1; i <= n; i = i * 2) ... log base 2 (n)
doIt(); ... log base 2 (n) * n^2
return 1; ... 1
}
All this have left me wondering, how do you really find out what cost is
what? I've been asking around, trying to understand but there is really no
one who can really explain this to me. I really wanna understand how do I
really determine the cost badly. Anyone can help me on this?