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# Big O Notation Question. Why is this O(N)?

Why is the following expression O(N) and not O(logN)?

```count = 0;
int i = N;
while (i > 1) {
count++;
i = i - 2;
}
i = N;
while (i > 1) {
count++;
i = i/2;
}
```

When N is 10:

```count-1:  5

count-2: 3

log(N): 3
```

When N is 100:

```count-1:  50

count-2: 6

log(N): 6
```

When N is 100,000:

```count-1:  50,000

count-2: 16

log(N): 16
```

The number of iteration for the first code is just half N. But for the second code, it must be O(log(N)). Subtracting 2 on every iteration would take at least half N to finish. But dividing by two on every iteration would take log(N) run.

If you are wondering how the maths work:

lets write an equation for how many times should you divide N to reach one:

1 = N/2^x

we are looking for x which tells us how many divisions we have made. So, lets rearrange the equation above:

2^x = N

x = log2(N) .... taking the logarithm of both sides.