I'm trying to figure out which is better asymptotic complexity, $O(\log{\frac{n}{p}})$ or $O\left(\frac{\log{n}}{\log{p}}\right)$. $p$ is the amount of parallelism (i.e. number of cores), and $n$ is the problem size. When I plot them in Grapher.app with any constant value for $p$, the first looks better:But when I try to work out the math it doesn't seem right:$$O(\log{\frac{n}{p}}) = O(\log{n} - \log{p})$$$O\left(\frac{\log{n}}{\log{p}}\right)$ seems like a better bound than the above since it divides by $\log{p}$ instead of just subtracting it. What am I missing?
Why is log(n/p) asymptotically less than log(n)/log(p)
asymptotics