Exponential Growth: The Andromeda Strain
Exponents and logarithms: inverse functions
UBC Calculus Online Course Notes

Doublings: The exponential function with base 2

On the last page, we studied how the process of doubling can lead to a very sharp increase after a few "cycles" (or doubling time periods) of growth. But what would we see if we made observations at more frequent intervals? How big would the population be? For example, we could think of the following experiment:

An E. coli culture is initiated with a drop of fluid containing 100 cells. The cells are allowed to grow and multiply for 2.5 hrs. The experiment is stopped, and a cell count is made. How many cells would be found in the culture at the conclusion of the experiment?

The only difficulty in answering this question stems from the fact that the time at which the experiment is concluded is not an exact multiple of the doubling time. We have to figure out what happens between the cycles of twenty minutes.

You might wonder what happens to the bacteria between the doublings. Do they all divide at once exactly every twenty minutes? In that case, we would think of this graph as a sequence of steps, with the heigths of each of the steps getting bigger and bigger.

Or do some cells divide sooner and others later so that the growth looks smooth? Although the first situation can occur (in specially designed cultures of cells in which the cell cycles are synchronized) it is rather rare in nature. It is more likely that the cells will divide at intervals which are not exactly synchronized. Even though each division will make the number of cells increase by an integer, the effect of each additional cell on the whole population will be very tiny. (Remember that after a few divisions, we will be dealing with thousands and then millions of cells, so that the axes we use to follow this graphically will hardly detect change by one or two cells.) This leads us to think that perhaps the process of growth need not be described as discrete jumps from one level to the next, but rather as a smooth transition. We might think about defining a new function that represents the growth more smoothly.

Defining a continuous function for powers of two

In defining this new function, remember that we understand fractional powers as well; for instance, $  2^{1/2} = \sqrt{2}  $ and so forth. In addition, we would like to preserve the property that $  (2^a)^b = 2^{ab} $ . This tells us what to expect at many points in between the integer powers. For example, $  2^{2/3} 
= (2^2)^{1/3}  = 4^{1/3} = \sqrt[3]{4}.  $

In the next graph, we show the function obtained by "connecting the dots". Now, for every value of time (expressed in multiples, though not necessarily integer multiples, of the doubling time), we can read off the smoothly adjusted value of the population.

The variable x is related to time

We have surreptitiously defined a new variable, x, which also represents time in some way. What is the connection between the clock time, t, and the new variable, x? Since $ x=t/\tau $ , x measures the number of doubling cycles the process undergoes rather than a more conventional unit like seconds or hours. If the ratio is an integer, x and n match exactly. But otherwise, x is not restricted to integer values: it is just a real positive number.

Aside: What about negative powers ?

In some cases, we would like to peer backwards in time. For instance, suppose that we know how many bacteria are present now, but we would like to know how many were present an hour ago. Since 3 doubling cycles have ocurred, the population is now $  2^3  $ larger than it was an hour ago. In other words, the population then was $  2^{-3}  $ what it is now.

For this reason, we would like to understand our function $ 
2^x  $ for negative powers as well. The following table tells us how to get started.

Power $ 2^0    $ $ 2^{-1} $ $ 2^{-2} $ $ 2^{-3} $ $ 2^{-4} $ $ 2^{-5} $
Value 1 1/2 1/4 1/8 1/16 1/32

When we include both positive and negative values on our graph, we get the following picture. A smooth curve has been used to connect the points once more, leading to the function $ y=2^x $ which we have now defined over the whole real line (i.e. for all real numbers x):

For people who can't see Java, it looks like this: 

		|   * 
		|            2^x 
		|   * 
		|     <-------connected dots 
		| * 
	    *	| 
      *		| 
-------------------------------- x