In general remember the increasing order of complexities:

constant divided by polynomial < constant < $logn$ < $(logn)^k$ < $n$ < $nlogn$ < $n(logn)^k$ < $n^k$ < $n^k(logn)^l$ < $n^{k+l}$ < $d^n$

Use this as a first level of analysis and eliminate some of the options.

When not to use logarithm method : if, after applying logarithm, you get some constant or a constant multiple/factor.

example: $n^2$ and $n^{100}$. If you apply logarithms, both are same complexity - though you know they are not.

If using 'sufficiently large numbers' - be judicious in choosing the large numbers. And don't stop with one trial. Perform the test with 2 sufficiently large numbers, and then take the ratio of the two trials for each function - to identify which grows the largest. I like to choose powers of 2, like $2^{1024}$, $2^{256}$ - since they'll also work if you have $loglogn$.

Unfortunately there is no one-size-fits-all approach. You need to use everything you know about function growth and intuition to reach the solution.