I'm comparing efficiencies for the famous fake-coin algorithms. Specifically, I'm looking at a two-pile approach and a three-pile approach for a solution. I have found that, like a binary search, the two-pile approach efficiency grows at the rate of $\log$(base $2$)$n$, while the three-pile approach efficiency grows at the rate of $\log$(base $3$)$n$.
So I want to compare the rates at which they grow for large values of $n$; I made a ratio of two-pile to three-pile growth as follows:
$\dfrac{\log_2 n}{\log_3 n}$
I want my answer to NOT depend on $n$. I actually know the answer, but I want to know what logarithmic rules, arithmetic, etc., are used to find the answer.
Here is the answer, step by step:
STEP 1. $\dfrac{\log_2 n}{\log_3 n}$
STEP 2. = $\dfrac{\log_2 n}{\log_3 2 \log_2 n}$
STEP 3. = $\log_2 3$
STEP 4. approximately $= 1.6$
Edit: I forgot to add that I set up a recurrence relation, prior to steps 1-4, that sets $n = 3^k$, so I don't know if that effects the answer, I don't think it does.