This page calculates standard times for growth or decay,
starting from data values given as a function of time, and assuming an
exponential relationship.
It provides references to sections and video answers in the book (e.g. 5.2.3,
Q5.27), to additional digital support
on this website (e.g.
Excel2007 ,
Mathematics of e, ln and log), or to other useful sources.
Main Topics:
Standard exponential
growth/decay equation
Analysing data recorded as population/quantity against
time
Using Excel to perform the calculations
Theoretical basis:
 Revision of powers and
indices
 Using logarithms to
rearrange exponential equations
 Exponential growth
and decay
 Linearisation of
exponential data
Download this webpage in pdf format

Any exponential growth or decay can be described by an equation of the form (see 5.2.3, Q5.21, Q5.22, Q5.23):
N_{t} = N_{0}e^{kt}
where
N_{t} is the population or quantity at time,
t,
N_{0} is the starting population or quantity
at time, t = 0,
k is the growth constant, which will be negative for a
decay.

Starting with values of the population/quantity, N_{t}, for different values of time, t:
Step 1. Plot a graph of ln(N_{t})
on the yaxis against t on the xaxis (see 5.2.6,
Q5.27).
(NB ‘ln’ is the logarithm to base ‘e’  see 5.1.5)
If you plot ‘log to base 10’, log(N_{t}), against t, then
you must use a different calculation for k  see in Step 3 below.
Step 2. Calculate the slope, m, of this graph (see 4.2.1 and 4.2.4, Q4.20)
Step 3. Derive the value for the growth constant, k (see 5.2.6, Q5.27):
k = m
(assuming that you plotted ln(N_{t}) against t)
or
k = 2.303×m (if you plotted log(N_{t})
against t)
Step 4. Calculate standard ‘times’ for growth/decay (see 5.2.4,
Table 5.1, Q5.26):
· Generation time: T_{G} = 0.693 / k
· Halflife: T_{1/2} =  0.693 / k
· Decay time constant: t = 1.000 / k
· Decimal reduction time: T_{D} = 2.303 / k

To use Excel to linearise and analyse exponential data 

 
see ‘Linearisation’ worksheet in ‘Logarithms and Exponentials’ for Excel2007 or Excel2003 

Revision of powers and indices
Check your understanding of powers and indices by 

 
attempting questions 16 to 22 in the Self Assessment Diagnostic Test and checking the Video Answers and Feedback 
Study Revision Notes for Powers 

Using logarithms to rearrange exponential equations
Check your understanding of logarithms by 

1. 
attempting questions 39 and 40 in the Self Assessment Diagnostic Test and checking the Video Answers and Feedback 
2.  attempting the selfassessment Mathematics of e, ln and log 
Study sections 5.1.1 to 5.1.6 

Check your understanding of exponential growth/decay 

 
attempting the selfassessment Exponential Growth and Decay 
Study sections 5.2.1 to 5.2.5 

Linearisation of exponential data
Study Example 5.4, section 5.2.6, and Example 5.18  see Q5.27

Comments and suggestions to: graham.currell@uwe.ac.uk