In the lag phase, the number of cells doesn't increase. However, considerable metabolic activity is occurring as the cells prepare to grow. The number of microbes in an exponentially increasing population increases slowly at first, then extremely rapidly. Organisms in a tube of culture medium can maintain log growth for only a limited time, as nutrients are used up, metabolic wastes accumulate, microobes suffer from oxygen depletion. In this phase, cells begin to die out.
Death occurs exponentially, but at a low rate.
Death occurs because cell have depleted intracellular ATP reserves. In the lab, cultures usually pass through all growth phases - not in nature. In nature, nutrients continuously enter the cell's environment at low concentrations, and populations grow continually at a low but steady rate. The growth rate is set by the concentration of the scarcest or limiting nutrient, not by the accumulation of metabolic byproducts - in nature there is always some other microbe that can use these metabolic byproducts for their own metabolism. In the lab, we have to continually replace the media.