How does $3n+1$ change the proximity of $n$ to a power of two?

Question

This is part of an attempt to prove Collatz's conjecture. I proved a modification of Collatz's conjecture, where instead of $3n+1$ if $n$ is odd, you do $n+1$. In Collatz's conjecture, if you get to a power of two, you're going to get to one. Let $p$ represent the proximity to a power of two. We shall keep the value of $p$ as $⌈\log_2(n)⌉-\log_2(n)$. Where $p=0$, $n$ must be a power of two. With my $n+1$ version, $p$ decreases if $n$ is odd, obviously and will stay the same if $n$ divides by two. However, is there a way to explain how $p$ decreases if $n$ becomes $3n+1$?