So I've been experimenting with sugar washes and Fleishmann's active dry yeast which comes in little square packages. When you open the package and look in, the yeast is in the form of a powder of nearly spherical nodules. I'm sure since yeast are microscopic, then each of these nodules must contain hundreds of thousands if not millions or billions of the fungal spores. And since I know yeast reproduce exponentially the proof should just need enough of the nodules to ensure there is a low probability that all the yeast in a given nodule is dead or something, and enough time for the exponential function to take the starting population to the desired population. So my theory is that if the yeast double every two hours, and I pitch half a package, then I should see the same activity after 2 h, 15 min as I saw after 15 min of pitching the whole package. Or if I pitch 1/4096'th of the package and come back one day later (because 24 hours is like 12 doubling periods so 2^12=4096) then again assuming the small amount of the dry yeast was alive you'd see the same amount of life right?? The only experiments I've done with this is to put smaller and smaller amounts of the package into the starting 1/4'th of a cup of warm sugar water, and the less you put the less it does, and after a few hours if I didn't put enough then I give up and add the rest of the active dry yeast powder.
Spray dried yeast is what the spherical nodules are composed of. Brewers yeast has lost the ability to form spores.
Yeast population do grow rapidly in the log phase of yeast growth. That is usually preceded by a lag phase immediately after pitching, where the yeast seems to be less active or even dormant. After some time (usually < 1 day) in the lag phase the yeast enters the log phase, where it can be observed to be rapidly budding. So the exponential growth does not always start on pitching the yeast in the wort/starter. The shortest lag phase and time to first budding can be obtained using larger pitches of yeast comprising the youngest parent cells. When the yeast has multiplied and is beginning to substantially reduce the nutrients in the wort it begins to enter a stable phase and yeast growth is reduced considerably, it certainly does not exhibit continuously exponential growth.
The strategy of adding a very small amount of yeast to a full brew is "not optimal". It can be done of course but the results may be more variable than desired. If the idea is to grow yeast on a to make several brews from an expensive yeast packet - then that might best be done in a small (say 1L) starter brew. When it has been fermenting for a day, it can be pitched - or split and one part pitched and the other part grown on. Yeast is generally cheap enough for home brewers that using a new packet every time is a viable and economic procedure.
It is best to pitch a "full volume" of yeast and get it fermenting in the wort as soon as is practically possible. A good guide is to use about 10g of dried yeast per 24L of wort. IMHO it is best to re-hydrate the yeast before adding it. The main problem with under pitching is that while the yeast is growing to full volume - other microbes can grow too. Yeast at full volume can metabolise wort sufficiently rapidly to inhibit other microbes. The alcohol produced is also a good bacterial inhibitor and beer stabiliser. It is a useful brewing objective to get the wort alcoholic and under a layer of CO2 as soon as is practically feasible. That generally isn't done by under pitching the amount of yeast.
There actually is a minimum amount you can pitch while still achieving sufficient attenuation. What it all comes down to is lipids, certain types of which are critical for yeast growth. Without enough, yeast can't create new cell membranes and therefore cannot reproduce and grow.
Without going into too much detail, it goes like this:
The initial yeast pitch will either already contain (in the case of oxidatively grown dry yeast), or will synthesize (in the case of liquid yeast or repitched slurries), a fixed amount of the needed lipids. Synthesis requires addition of dissolved oxygen.
Upon cell division the available lipids are divided between mother and daughter cell. This continues on as long as the yeast sense the available lipids are sufficient for another round of division. In essence, each round of division will halve the amount of lipid left in the total yeast culture. So, as there is an exponential increase in cell number, there is also an exponential decrease in the lipid pool.
As soon as the yeast sense there is insufficient lipid for another round of division, yeast growth (i.e. the increase in the number of cells) will cease. This happens regardless of whether all other nutrients are still sufficient. At this point, yeast begin preparation for a hibernation phase. If this happens before the required attenuation is reached, you've got what's called a stuck or hung fermentation.
In your specific case, the maximum possible cell count is going to be limited by the amount of lipids contained in or synthesized by a very small initial amount of yeast. Unless you oxygenate further, the available lipids will rapidly be diluted to the limiting point. Basically the smaller the initial pitch, the smaller the initial lipid pool, and thus the lower the terminal cell count possible.
Your general understanding of logarithmic growth seems accurate, but assumes unlimited supplies of all critical nutrients, which is not the case. When using pitching rates (1-2 million cells per mL per degree Plato) and oxygenation rates (~10 ppm) typical in brewing, there are generally no more than 4 doublings of the yeast. Therefore I'd expect that, regardless of how much you pitch, you shouldn't expect to see more doublings than this, not at least without adding more oxygen. I won't bother to guess at the minimum amount of dry yeast needed in your situation as this will depend on so many things, but I hope this gives you an idea of the reasons behind it.
Proper pitch amounts are well documented. What effects the amount you need is the OG and volume of the wort. Most dry yeast packs are a proper pitch for 5 gallons of 1.040 OG wort. Each dry pack has about 200 billion cells.
Yes you can grow a lot of yeast from very little. As to your question: All that's needed is a single cell. That would be the minimum "needed". But pitching very little yeast in a batch of you want to ferment as a drinkable product will have a lot of bad side effects.
There is much much more to this topic. Here's a couple things for you to reasearch.
- Yeast pitch calculators.
- Yeast slants
- Yeast starters
- Underpitching yeast
- Yeast growth phase esters
- Multi-step yeast starters
- Yeast life cycle
It is really not as easy as you describe it. Yeast does not have a single phase where it only consumes sugar and produces CO2 and alcohol, and grows indefinitely.
When you pitch it in a wash (or wort, or anything containing sugar), it will use up dissolved oxygen to grow, and depending upon other factors you will get exponential/large growth. However, once this oxygen is used up, the yeast will still grow, but much slower.
The only way that you can double yeast every couple of hours is by building a yeast starter, and oxygenating it using a stir plate. However, even then there is a limit to its growth. That is why to build a really big starter, the yeast will be grown in several phases, where in every phase a bigger amount of wort (in case of beer starters) is used.