exponential phase (sometimes called the log phase or the logarithmic phase) is a period characterized by cell doubling.
Three species of bacteria, Carnobacterium pleistocenium, Chryseobacterium greenlandensis, and Herminiimonas glaciei, have reportedly been revived after surviving for thousands of years frozen in ice. As a practical matter, as a researcher, you will want to preserve your selected bacteria so you can go back to it if something goes wrong.
Whenever you successfully transform a bacterial culture with a plasmid or whenever you obtain a new bacterial strain, you will want to make a long term stock of that bacteria. Bacteria can be stored for months and years if they are stored at -80C and in a high percentage of glycerol.
In order to ensure a pure culture is being preserved, pick a single colony of the bacteria off a plate, grow it overnight in the appropriate liquid media, and with shaking . Take the overnight culture and and mix an aliquot with 40% glycerol in sterile water and place in a cryogenic vial. It is important to label the vial with all the relevant information (e.g. strain, vector, date, researcher, etc.). Freeze the glycerol stock and store at -80C. At this point you should also record the strain information and record the location.
While it is possible to make a long term stock from cells in the stationary phase, ideally your culture should be in logarithmic growth phase. Certain antibiotics in the medium should be removed first as they are supposedly toxic over time, e.g. Tetracycline. To do this, spin the culture down and resuspend it in the same volume of straight LB medium.