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IBMaths: AI HLDPRevision Notes4. Statistics & Probability4.3 Further Correlation & Regression4.3.2 Logarithmic Scales

Logarithmic Scales (DP IB Maths: AI HL)

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Dan

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Dan

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Logarithmic Scales

What are logarithmic scales?

  • Logarithmic scales are scales where intervals increase exponentially
    • A normal scale might go 1, 2, 3, 4, ...
    • A logarithmic scale might go 1, 10, 100, 1000, ...
  • Sometimes we can keep the scales with constant intervals by changing the variables
    • If the values of x increase exponentially: 1, 10, 100, 1000, ...
    • Then you can use the variable log x instead which will have the scale: 1, 2, 3, 4, ...
    • This will change the shape of the graph
      • If the graph transforms to a straight line then it is easier to analyse
  • Any base can be used for logarithmic scales
    • The most common bases are 10 and e

Why do we use logarithmic scales?

  • For variables that have a large range it can be difficult to plot on one graph
    • Especially when a lot of the values are clustered in one region
    • For example: populations of countries
      • This can range from 800 to 1 450 000 000
  • If we are interested in the rate of growth of a variable rather than the actual values then a logarithmic scale is useful

log-log & semi-log Graphs

What is a log-log graph?

  • A log-log graph is used when both scales of the original graph are logarithmic
    • You transform both variables by taking logarithms of the values
  • log y & log x will be used instead of y & x
  • Power graphs (space y equals a x to the power of b) look like straight lines on log-log graphs

What is a semi-log graph?

  • A semi-log graph is used when only one scale (the y-axis) of the original graph is logarithmic
    • You transform only the y-variable by taking logarithms of those values
  • log y  will be used instead of y
  • Exponential graphs (space y equals a b to the power of x) look like straight lines on semi-log graphs

How can I estimate values using log-log and semi-log graphs?

  • Identify whether one or both of the scales are logarithmic
  • Identify the variable so that the scales have equal intervals
    • x : 1, 10, 100, 1000, ... use log x
    • For x : 1, e, e², e3, ... use ln x
  • If you are asked to estimate a value:
    • First find the value of any logarithms
      • For example: log y, ln x, etc
    • Use the graph to read off the value
    • If it is a value for a logarithm find the actual value using:
      • log invisible function application x equals k rightwards double arrow x equals 10 to the power of k
      • ln invisible function application x equals k rightwards double arrow x equals straight e to the power of k

Examiner Tip

  • Pay close attention to which base is being used (log or ln)

Worked example

The function space y equals f left parenthesis x right parenthesis is drawn below using a log-log graph.

2-7-1-we-imageShow that when x equals 56 the value of space y is approximately 24.

2-7-1-ib-ai-hl-log-log-graphs-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.