DP IB Maths: AI HL

Revision Notes

IBMaths: AI HLDPRevision Notes4. Statistics & Probability4.3 Further Correlation & Regression4.3.3 Linearising using Logarithms

4.3.3 Linearising using Logarithms

Test Yourself

Exponential Relationships

How do I use logarithms to linearise exponential relationships?

  • Graphs of exponential functions appear as straight lines on semi-log graphs
  • Suppose y equals a b to the power of x
    • You can take logarithms of both sides
      • ln y equals ln left parenthesis a b to the power of x right parenthesis
    • You can split the right hand side into the sum of two logarithms
      • ln y equals ln a plus ln left parenthesis b to the power of x right parenthesis
    • You can bring down the power in the final term
      • ln y equals ln a plus x ln b
  • ln y equals ln a plus x ln b is in linear form Y equals m X plus c
    • Y equals ln y
    • X equals x
    • m equals ln b
    • c equals ln a

How can I use linearised data to find the values of the parameters in an exponential model y = abx?

  • STEP 1: Linearise the data using Y equals ln y and X equals x
  • STEP 2: Find the equation of the regression line of Y on X : Y equals m X plus c
  • STEP 3: Equate coefficients between Y equals m X plus c and ln y equals ln a plus x ln b
    • m equals ln b
    • c equals ln a
  • STEP 4: Solve to find a and b
    • a equals straight e to the power of c
    • b equals straight e to the power of m

Worked example

Hatter has noticed that over the past 50 years there seems to be fewer hatmakers in London. He also knows that global temperatures have been rising over the same time period. He decides to see if there could be any correlation, so he collects data on the number of hatmakers and the global mean temperatures from the past 50 years and records the information in the graph below.

2-7-2-a-we-image

Hatter suggests that the equation for h in terms of t can be written in the form h equals a b to the power of t
. He linearises the data using x equals t  and y equals ln h and calculates the regression line of y on x to be y equals 4.382 minus 1.005 x.

 Find the values of a and b.

2-7-2-ib-ai-hl-exp-relationships-we-solution

Power Relationships

How do I use logarithms to linearise power relationships?

  • Graphs of power functions appear as straight lines on log-log graphs
  • Suppose y equals a x to the power of b
    • You can take logarithms of both sides
      • ln y equals ln left parenthesis a x to the power of b right parenthesis
    • You can split the right hand side into the sum of two logarithms
      • ln y equals ln a plus ln left parenthesis x to the power of b right parenthesis
    • You can bring down the power in the final term
      • ln y equals ln a plus b ln x
  • ln y equals ln a plus b ln x is in linear form Y equals m X plus c
    • Y equals ln y
    • X equals ln x
    • m equals b
    • c equals ln a

How can I use linearised data to find the values of the parameters in an power model y = axb?

  • STEP 1: Linearise the data using Y equals ln y and X equals ln x
  • STEP 2: Find the equation of the regression line of Y on X : Y equals m X plus c
  • STEP 3: Equate coefficients between Y equals m X plus c and ln y equals ln a plus b ln x
    • m equals b
    • c equals ln a
  • STEP 4: Solve to find a and b
    • a equals straight e to the power of c
    • b equals m

Worked example

The graph below shows the heights, h metres, and the amount of time spent sleeping, t hours, of a group of young giraffes. It is believed the data can be modelled using t equals a h to the power of b
.

2-7-2-b-we-image

The data are coded using the changes of variables x equals ln h and y equals ln t. The regression line of y on x is found to be y equals 0.3 minus 1.2 x.

Find the values of a and b.

2-7-2-ib-ai-hl-power-relationships-we-solution

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    Dan

    Author: Dan

    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.