Is the Drake equation based on a logical fallacy? I often wonder why the Drake equation gets taken so seriously and why popular scientists in general say "it's probable' life exists elsewhere in the cosmos.
My problem is that the mechanism for abiogenesis is as yet unknown. 

We don't know the likelihood that RNA like molecules can assemble from chance or for that matter that this is the source of life on this planet (as opposed to coming via a comet). We do know that we have no record of it recurring and as yet Miller-uri style experiments have not yielded results.

The argument I keep hearing from Dawkins, Sagan and many others is that the universe is vast and life occurred relatively quickly in our planet's history which suggests it is common.
I detect a logical fallacy based on the inductive reasoning of using one occurrence in order to establish a pattern. 

Think of it this way. If I've never played cards and get dealt my first hand which is a royal flush I might think its quite likely based on my past hand that I will get another in the next deal. This doesn't change if the deck contains a hundred suites and the odds of a royal flush is astronomical - if my first hand is a royal flush.

Similarly the vastness of the cosmos tells you nothing about the likelihood of other life if you don't the probability of its occurrence.It could be very likely or so rare as to only occur once. 
Does anybody have an opinion on this?
To directly answer the question, the Drake Equation is not based on a "logical fallacy" (which means a different thing, philosophically). The logic is fine, the real problem is we don't have any meaningful data.

Regarding abiogenesis (the first formation of some simple life): as you correctly state, we know only that it happened once (here, presumably but that doesn't matter). However the argument that it happened quite quickly on our planet is meaningless statistically, because that's just a sample of one.

(If anyone thinks this answer is too waffly or off topic just say so and I'll delete it by the way; I think the question is a fair one but we just can't present any data, just reason.)