EXPONENTIAL GROWTH

Be sure you are acquainted with the two forms of the equations for exponential growth and decay. Recall that they are:

When introducing the equations, we mentioned a case of wee beasties. There were 10% increases in the population. One population began with a population of 100, and after a year, there were 110. The other population had a population of 5000, and one year later, it grew to 5500. Note the ratios of final to initial populations, 'N/', were both the same:

As you can see, for a one year interval, this ratio was 1.1.

You can solve for 'k', the growth constant, for this particular example using the second equation. Since N/ = 1.1, and t = 1.0 (year), we have

ln (1.1) = k (1.0)
Since ln (1.1) = 0.09531, then k = 0.0953/year.

Recall that an exponent must be dimensionless. So 'k' will always have dimensions of reciprocal time. In the case of the wee beasties, k has units of year.

Now that we know the value of the growth constant for our wee beasties, k = 0.0953, we can substitute this into our first equation.

Suppose the initial population were 2000 and we wish to calculate what it will be in 5.5 years. We know 'k' (= 0.0953 year), 't' (= 5.5 year), and '' (= 2000). Then we can calculate 'N':
Note that if the growth constant 'k' were larger, then 'kt' would be larger at any given time, and so the increase in population would be greater.

Exponential Decay