Logarithmic and Exponential Functions
5.2 Exponential Growth and Decay
5.2.1
Introduction
A growth (or decay) is said to be exponential if time appears as an exponent or power in the equation of growth.
There are many examples of exponential growth or decay systems in all branches of science, and different disciplines have devised different ways of quantifying very similar processes, e.g. half-life in radioactivity, the elimination constant for drug concentration and the amplification of a photomultiplier. However, in this unit we see how many simple growth (or decay) systems can be represented, and analysed, by using one common exponential equation using the base e.
5.2.2
Exponential systems
There are many systems in science where the future change in the system depends on the
current state of the system. For example:
• A population (e.g. bacterial cells in blood system) may increase by a constant proportion of the current population.
• In radioactive decay, the rate of decrease in radioactivity is proportional to the current level of radioactivity.
Q5.13
A poor (but consistent) gambler loses exactly half of his remaining money, M,
every week. He starts withM0 = £640 at the beginning (n = 0) of the first week, and is down to M1= £320 at the end of the first week (n = 1) and down to
M2= £160 at the end of the second week (n = 2). For each value ofn in the table below calculate:
(i) how much money,Mn, is left by halving the previous value (ii) the value ofMn, using the equationM= 640 × 0.5n.
End of weekn= (0) 1 2 3 4 5 6 7
(i) Money,Mn(£)= 640 320 160
(ii) Mn= 640 × 0.5n= 640
The answers to Q5.13 show that a proportionate change can be mathematically represented by an exponential equation. The exponential decay for Q5.13 is reproduced in Figure 5.1.
0 160 320 480 640 0 1 2 3 4 5 6 7 Number of weeks, n 1 week Mn 2 weeks 3 weeks
Figure 5.1 Exponential decay:Mn= 640 × 0.5n.
If a population increases by a factorg within a given time period T , then
Growth factor, g= Population at end of period, T
Population at start of period, T
Example 5.11
Calculate the growth factors, g, in the following situations:
(i) a population increases by 5 % every 10 years (ii) a bacterial population decreases by 20 % every hour. (i) Period,T = 10 years: g = 100%+ 5%
100% =
105%
100% = 1.05
(ii) Period,T = 1 hour: g = 100%− 20%
100% =
80%
100% = 0.8
A value ofg < 1 shows a decaying population.
Q5.14
Calculate the growth factors, g, in the following situations:
(i) A population doubles.
(ii) A population increases by 50 %. (iii) A population falls by 10 %. (iv) A population falls to 10 %.
When describing the growth (or decay) of a population we can write that: Population at the start (n= 0) is given by N0.
Population aftern periods is given by Nn.
The sequence of populations below for each period of a growth shows that the population increases by a factorg every period, giving N0, N0× g, N0× g2, N0× g3, etc.
Period Change in population Population 0 N0 (starting population) → N0 1 N1= N0× g equivalent to: → N1 = N0× g1 2 N2= N1× g substituting forN1: N2= (N0× g) × g → N2 = N0× g2 3 N3= N2× g substituting forN2: N3= (N0× g2)× g → N3 = N0× g3
n giving, aftern steps: Nn= (N0× gn−1)× g → Nn=N0×gn
Hence, aftern periods, the population, Nn, is described by an exponential equation:
Nn= N0× gn [5.21]
Example 5.12
What are the values for the period T (in days) and the factor g in the example given
The question states that after 1 week the money remaining has fallen to one-half. Hence g= 0.5 for a period T = 7 days.
In general, the time,t, taken for n periods of duration T will be given by
t = n × T and n= t
T [5.22]
Hence we can rewrite equation [5.21] directly in terms oft instead of n.
A population,Nt, which grows by a factorg in a time T is given by:
Nt = N0× gt/T [5.23]
Figure 5.2 shows the exponential decay of an initial population of N0= 640, which has a growth factor ofg= 0.5 over a time period, T = 7 days (= 1 week). This is the same decay
as in Figure 5.1, but the time,t, is now expressed in terms of days using equation [5.23].
0 160 320 480 640 0 10 20 30 40 50 Time, t /days 1 week Nt 2 weeks 3 weeks
Figure 5.2 Exponential decay:Nt= 640 × 0.5t/T withT = 7 days.
The equation,Nt = N0× gt/T, can be used to describe the growth (decay) of any exponential
system over a time,t.
For a particular system it would be necessary to choose appropriate values for each of the variables,N0, g and T .
Example 5.13
The equation for the growth of bacteria (see 5.2.4) can be written as Nt = N0× 2t/TG, whereTGis the generation time of the bacteria.
In this case the bacteria numbers will double (growth factor,g= 2) within the period, TG= generation time.
Q5.15
In a ‘chain letter’, one person sends a letter to six people who are each ‘encouraged’ to forward the letter to six new people within 2 weeks.
Assume that the chain is not broken and all six people actually forward the letter to new recipients.
(i) What is the ‘period’ of change in this example? (ii) What is the value forg in this example?
(iii) Using an equation of the form, Nt= N0× gt/T, estimate how many weeks, months or years it would take before at least 20 million people have become involved in the chain.
Q5.16
A bacterial culture in the ‘death phase’ has an initial population of 2.0× 106 cells per mL, which decays exponentially to one-tenth of its initial population in 1.2 hours.
(i) Calculate the population after 3.6 hours. (ii) What is the value forg in this example?
(iii) Using an equation of the form,Nt = N0× gt/T, calculate the population after 2.2 hours.
5.2.3
Exponential growth equation N
t= N
0e
ktGrowth and decay in different areas of science have produced different expressions of the basic equationNt = N0× gt/T. However, the problem with this form is that different examples may need different values for the base,g.
It is more convenient to use a standard equation that always uses the same ‘base’. For reasons that may not be obvious at first, the best choice for a common ‘base’ is Euler’s number, e:
(6.2.2) of growth and decay equations.
In the above equation, we replaced the growth factor,g, and time period, T , with a single
exponential growth factor,k, whose value will depend on the value of g. Since the product, k× t, is a pure number, k will have units of ‘1/time’.
Comparing equations [5.23] and [5.24], the two equations will be identical if:
ekt= gt/T
Taking natural logarithms of both sides:
ln(ekt)= ln
gt/T
Using equation [5.8], the above equation becomes:
kt = t
T × ln(g)
Cancellingt from both sides, we get:
k= ln(g)
T and thenT =
ln(g)
k [5.25]
5.2.4
Specific applications of N
t= N
0e
ktWe identify four examples of using specific time periods in growth and decay:
1. The exponential growth of a bacterial population is often measured by the generation time,