Using Exponentials & Logarithms in Modelling (Cambridge (CIE) A Level Maths) : Revision Note

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Using Exps & Logs in Modelling

What is exponential modelling?

  • Exponential modelling is where real-life situations are modelled using exponential growth and decay 

  • Exponential growth is typically used for …

    • … population (animals and humans) increase

    • … the value of an investment under compound interest

Using Exps & Logs in Modelling Notes fig2, A Level & AS Maths: Pure revision notes
  •  Exponential decay is typically used for …

    • … levels of radioactivity

    • … quantity of a drug in a person’s bloodstream

    • … depreciation (eg value of a standard car)

Using Exps & Logs in Modelling Notes fig1, A Level & AS Maths: Pure revision notes
  • It is important to consider if a model is only relevant for a certain period of time

  • Can a population increase forever?

    • Will a very old car still be worth a small amount of money?

  • For example: Consider a substance that initially contains 2000 radioactive atoms where the number of radioactive atoms, N, at time, t hours, is modelled by N equals 2000 e to the power of negative k t end exponent

    • y is N

    • A = 2000 – the initial number of radioactive atoms

    • k > 0 so the negative indicates this must be exponential decay

    • t is used (rather than x) as the situation is time-dependent

How do I solve modelling problems? 

  • Recognise the form of an exponential model

    • Growth y = Aekt

    • Decay y = Ae-kt

  • A (constant) is the starting or initial value

    • When t = 0,

      ekt = e0 = 1

      so   y = A

  • k (constant) determines the rate of growth/decay

  • Many modelling problems are about rates of change

    • If y = Aekt

      then     dy/dt = Akekt

    • If y = Ae-kt

      then     dy/dt = -Ake-kt

Using Exps & Logs in Modelling Notes fig4, A Level & AS Maths: Pure revision notes
Using Exps & Logs in Modelling Notes fig5, A Level & AS Maths: Pure revision notes
Using Exps & Logs in Modelling Notes fig6, A Level & AS Maths: Pure revision notes

Worked Example

Using Exps & Logs in Modelling Example fig1, A Level & AS Maths: Pure revision notes
Using Exps & Logs in Modelling Example fig2, A Level & AS Maths: Pure revision notes
Using Exps & Logs in Modelling Example fig3, A Level & AS Maths: Pure revision notes
Using Exps & Logs in Modelling Example fig4, A Level & AS Maths: Pure revision notes
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Paul

Author: Paul

Expertise: Maths Content Creator (Previous)

Paul has taught mathematics for 20 years and has been an examiner for Edexcel for over a decade. GCSE, A level, pure, mechanics, statistics, discrete – if it’s in a Maths exam, Paul will know about it. Paul is a passionate fan of clear and colourful notes with fascinating diagrams.

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