Natural Logarithmic Models
What are the parameters of natural logarithmic models?
- A natural logarithmic model is of the form
- 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, , in decibels (dB) can be modelled by the function
,
where 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 .
b)
Find the sound intensity, in Wm-2, of a car alarm that has a sound intensity level of 105 dB.