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

Logarithmic functions & graphs

What are the key features of logarithmic graphs?

• A logarithmic function is of the form 

• The graphs always pass through the point

• The graphs do not have a y-intercept

  • The graphs have a vertical asymptote at the y-axis

• The graphs have one root at 

  • This can also be found using your GDC

• The graphs do not have any minimum or maximum points

• The graphs are monotonic

  • The value of determines whether the graph is increasing or decreasing

    • If is positive then the graph is increasing

    • If is negative then the graph is decreasing

Natural logarithmic models

What are the parameters of natural logarithmic models?

• A natural logarithmic model is of the form 

• The value of represents the value of the function when

• The value of 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

  • In real-life this might not be the case

    • The variable might have a limit

• The rate of change varies rapidly initially

  • In real-life it might change slowly and then speed up