## Study Plan:     Analysis of Exponential Growth/Decay using Logarithms

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:

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### Standard exponential growth/decay equation

Any exponential growth or decay can be described by an equation of the form (see 5.2.3, Q5.21, Q5.22, Q5.23):

Nt = N0ekt

where
Nt is the population or quantity at time, t,
N0 is the starting population or quantity at time, t = 0,
k is the growth constant, which will be negative for a decay.

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### Analysing data recorded as population/quantity against time

Starting with values of the population/quantity, Nt, for different values of time, t

Step 1. Plot a graph of ln(Nt) on the y-axis against t on the x-axis (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(Nt), 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(Nt) against t)
or
k
= 2.303×m          (if you plotted log(Nt) against t)

Step 4. Calculate standard ‘times’ for growth/decay (see 5.2.4, Table 5.1, Q5.26):

·         Generation time: TG = 0.693 / k
·
Half-life: T1/2 = - 0.693 / k
·
Decay time constant: t = -1.000 / k
·
Decimal reduction time: TD = -2.303 / k

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### Using Excel to perform the calculations

 To use Excel to linearise and analyse exponential data - see ‘Linearisation’ worksheet in ‘Logarithms and Exponentials’ for  Excel2007 or Excel2003

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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

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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 self-assessment Mathematics of e, ln and log Study sections 5.1.1 to 5.1.6

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Exponential growth and decay

 Check your understanding of exponential growth/decay - attempting the self-assessment Exponential Growth and Decay Study sections 5.2.1 to 5.2.5

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Linearisation of exponential data

Study Example 5.4, section 5.2.6, and Example 5.18 - see Q5.27

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