Then
If we are given an initial amount y, say y 0 at time t = 0, then
y 0 = c · e k · 0 = c · 1 = c,and our law of exponential change
y = cekt (2)tells us that c is the initial amount of y (at time t = 0). If the quantity grows with
time, then k > 0; if it decays (or diminishes, or decomposes), then k < 0.
Equation (2) is often referred to as the law of exponential growth or decay.
Half-lifeThe length of time required for a quantity that is decaying exponentially to be
reduced by half is called its half-life.
Example 12 __
The population of a country is growing at a rate proportional to its population. If
the growth rate per year is 4% of the current population, how long will it take for
the population to double?
SOLUTION:
If the population at time t is P, then we are given that . Substituting in
equation (2), we see that the solution is
P = P 0 e0.04t,where P 0 is the initial population. We seek t when P = 2P 0 :