Are the following Big Oh Notations equivalent?
This kind of thing just doesn't work. For example, one of your intermediate terms is $O(1)/O(\log n)$. However any function&nbsp;$f$ can be written as $g/h$ where $g=O(1)$ and $h=O(\log n)$.  If $f=O(1)$, then let $g=f$ and $h=1$. If $f=\Omega(1)$, then let $g=1$ and $h=1/f$.

For a more formal treatment, try to rewrite each of your statements that involve big-$O$ in terms of "There are constants $k$ and $n_0$ such that, for all $n\geq n_0$, $f(n)\leq k\,g(n)$." You'll probably find that it's not even clear what these formal statements should be and that, if you manage to figure that out, then some of the formal statements will be false.

The underlying point you've overlooked is that big-$O$ provides upper bounds but, if you have a quotient $a/b$ then upper-bounding that quotient requires upper-bounding&nbsp;$a$ and lower-bounding&nbsp;$b$. Big-$O$ can't express that lower bound.