4.8
Exponential, Logarithmic and
Logistic Curve Fitting
Curve Fitting:
We have used our graphing calculators to
create models using scatter plot data.
So far, our models have looked linear,
quadratic and cubic.
We can also fit our data to models that are exponential, logarithmic, power
and logistic.
Your graphing calculator has options for all of these.
Logistic Model:
A logistic growth model is an
exponential function in which
the dependent variable is
limited.
c
P(t ) 
bt
1  ae
a, b, and c are constants
with c>0 and b>0
c is called the carrying capacity because the value P(t) approaches
c as t approaches infinity. c represents the maximum value the
function can attain.
Some examples of logistic models are population growth and the
growth of a flower.
Logistic Model:
Growth slows as the population reaches the carrying capacity of its environment.
4.9
Logarithmic Scales
Loudness of Sound
L( x)  10 log
x
I0
L(x) is the loudness, measured in decibels
x is the sound intensity, measured in watts per square meter
I 0  1012 watt per square meter is the least intense sound that a human ear can
detect.
I0
when x  I 0 , L( x)  10 log  10 log 1  0
I0
Thus at the threshold of human hearing, the loudness is 0 decibals. We
can use this fact to compare to the loudness of common sounds.
Loudness of Sound p386 #1
Find the loudness of a dishwasher that operates at an intensity of 10
watt per square meter. Express your answer in decibals.
5
L(10 )  10 log
105
I0
5
5
10
7
 10 log 12  10 log 10
10
= 70 decibals
Intensity of Sound:
If the loudness of a diesel engine is 90 decibels, what is
its sound intensity in watts per square meter?
The sound intensity of a diesel engine
with a loudness of 90 decibels is
watt per square meter.
10 5
 x
90  10 log  
 I0 
 x
9  log  
 I0 
x
10  12
10
109 10 12  x
9
10 3  x
Magnitude of an Earthquake
Richter Scale: A way of converting seismographic readings into numbers that
provide an easy reference for measuring the magnitude of an earthquake.
This number quantifies the energy released by an earthquake.
The Richter scale is a base-10 logarithmic scale which defines magnitude as
the logarithm of the ratio of the seismographic reading of a given earthquake
to a zero-level earthquake whose seismographic reading measures .001
millimeter. Both readings are taken from a distance of 100 kilometers from
the epicenter of the earthquakes.
 x
M ( x)  log 
 x0 
Where M(x) is the magnitude, or the Richter Scale number.
x is the seismographic reading of the earthquake in question.
x0  103
is the seismographic reading of a zero-level earthquake at a
distance of 100 kilometers from the epicenter of the
earthquake.
Magnitude of an Earthquake:
Find the magnitude of an earthquake whose seismographic reading is
15,848 millimeters at a distance of 100 kilometers from its epicenter.
15,848
M (15,848)  log
 7.2
3
10
This earthquake measured 7.2 on the Richter Scale.
Comparing Earthquakes: p391 #70
On September 9, 1985, the western suburbs of Chicago experienced a mild
earthquake that registered 3.0 on the Richter Scale. How did this earthquake
compare in intensity to the great San Francisco earthquake of 1906, which
registered 6.9 on the Richter Scale?
Let x1 represent the seismographic reading of the Chicago earthquake, x2
represent the seismographic reading of the San Francisco earthquake.
x1
3.0  log ,
x0
x1  10  x0
3
x2
6.9  log ,
x0
x2  106.9  x0
x1 103.0  x0 103.0
 6.9
 6.9  103.9  .0001259
x2 10  x0 10
The Chicago earthquake was .0001259 times as intense as the San Francisco
earthquake.