Please clarify for me: the science says to refrigerate at end of exponential phase rather than when fully fermented. Exponential phase generally over 18-24 hours after pitching into starter? Or 18-24 hours after first signs of activity in the starter? What am I looking for to know that exponential phase is over? What changes will I see?

If we assume that a White Labs vial contains 100 billion viable cells when shipped, a relatively fresh vial contains 50 billion viable cells, the maximum cell densities for 1L and 2L starters are 200 billion cells and 400 billion cells respectively, and yeast cells bud every ninety minutes after the lag phase has been exited, then we are looking at two and three propagation periods for 1L and 2L starters under perfect conditions. Adding one ninety minute propagation period to each count to account for cell death during propagation results in three cell division periods for a 1L starter and four cell division periods for a 2L starter.

propagation_time_for_a_1L_starter = lag_time_in_hours + (3 x 90 / 60) = lag_time_in_hours + 4.5

propagation_time_for_a_2L_starter = lag_time_in_hours + (4 x 90 / 60) = lag_time_in_hours + 6

The reason why the exponential phase (a.k.a. log phase) is called the exponential phase is because the cell count grows are a rate of 2

^{n}, where n is the number of 90 minute time periods that have elapsed since the end of the lag phase.

With that said, let's calculate how long it takes 200 billion cells to reach maximum cell density in a 5-gallon batch of wort. Five gallons is roughly 19 liters (19,000 milliliters); hence, the cell count from a 200 billion cell 1L starter has to increase by a factor of 19. Now, we are dealing with exponential, not linear growth; hence, the number of 90 minute time periods that are required to increase the cell count 19 fold is equal to log

_{2}(19), where log

_{2} is the log base 2 function. Most calculators do not support log

_{2}, but we can take the log

_{2} of n by taking the log(n) over the log(2) (i.e., log(n)/log(2)); hence, log(19) / log(2) = 5 (rounded). Five ninety minute replication periods after the lag phase has been exited should be enough time to reach maximum cell density.

Now, anyone who is following this thread closely has probably figured out that the phase over which we have the most control in a fermentation is the time spent in the lag phase. By stepping a culture at the end of the exponential phase, we are pitching yeast cells that require very little in the way of replenishment. Hence, we will experience a shorter lag phase than we will if we pitch a culture that has reached quiescence because a quiescent culture has to undo the morphological changes it underwent in preparation for hard times. A quiescent culture also has to replenish the ergosterol and UFA reserves that were spent post-exponential phase, which increases dissolved O2 requirements.

Here's another thing to think about when propagating a culture. Bacteria cells divide every thirty minutes on average. Hence, the bacteria cell count grows by a factor of 8 every time the yeast cell count grows by a factor of 2. Add in the possibility of a shorter lag phase for house bacteria, and it should be painfully obvious why pitching at the end of the exponential phase is preferred to waiting until the culture has entered quiescence.