Runtime-Analysis for single loop incremented by a factor of 3 So, I'm trying to understand how to  get the run time of this loop:

for(int i = 1; i &lt; n*n*n; i*=3) {...}


So, far I know:


loop starts at 1
finishes when $i  &gt; n^3-1$
that $i$ is multiplied by 3 each iteration. 


Intuitively it seems like as $n$ gets larger the number of iterations decreases which makes me think that the run-time is $O(log$ $n)$. My question is if this is correct if it is what is a mathematical way I could derive that answer?
That's right. This for loop stops when: 
$$
3^i &lt; n^3 \rightarrow i&lt;3\log(n)
$$
Which implies the complexity of this for loop is $O(\log(n))$.