I know that the height of a $d$-ary heap on $n$ nodes is $\lceil (\log_d (n(d-1) + 1) - 1)\rceil$, but I was wondering how to justify that that's $\Theta(\log_d n)$? I know the definition of $\Theta, O, \Omega$ if only one variable is involved. In particular, what is the formal definition of a function $h(n,d)$ being in $\Theta(\log_d n)$?
One can clearly upper bound the height by $\log_d (n)+1$, but why does that imply the height is in $O(\log_d n)$? The problem is that $d$ can exceed $n$, and if $d$ keeps increasing while $n$ is fixed, then $\log_d n$ will approach $0$.
Also, one can show that the height is at least $\log_d (n(d-1) + 1) - 1\ge \log_d n - 1$ for $d$ sufficiently large. Why is this in $\Omega(\log_d n)$?