Logarithmic Scales (DP IB Applications & Interpretation (AI)): Revision Note

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 into 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 () 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 () 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:

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