IBMaths: AI HLDPRevision Notes2. Functions2.6 Further Modelling with Functions2.6.2 Natural Logarithmic Models

Natural Logarithmic Models (DP IB Maths: AI HL)

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Natural Logarithmic Models

What are the parameters of natural logarithmic models?

  • A natural logarithmic model is of the form space f left parenthesis x right parenthesis equals a plus b space ln x
  • The a represents the value of the function when x = 1
  • The b determines the rate of change of the function
    • A bigger absolute value of b leads to a faster rate of change

What can be modelled as a natural logarithmic model?

  • A natural logarithmic model can be used when the variable increases rapidly for a period followed by a much slower rate of increase with no limiting value
    • M(I) is the magnitude of an earthquake with an intensity of I
    • d(I) is the decibels measured of a noise with an intensity of I

What are possible limitations a natural logarithmic model?

  • A natural logarithmic graph is unbounded
    • However in real-life the variable might have a limiting value

Worked example

The sound intensity level, L, in decibels (dB) can be modelled by the function

L left parenthesis I right parenthesis equals a plus 8 ln I,

where I is the sound intensity, in watts per square metre (Wm-2).

a)
Given that a sound intensity of 1 Wm-2 produces a sound intensity level of 110 dB, write down the value of a.

2-6-2-ib-ai-hl-natural-log-model-a-we-solution

b)
Find the sound intensity, in Wm-2, of a car alarm that has a sound intensity level of 105 dB.

2-6-2-ib-ai-hl-natural-log-model-b-we-solution

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    Dan

    Author: Dan

    Expertise: Maths

    Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.