Chapter 1.3
Chapter 1.3 Log Models 2
The logarithmic model has the following form:
, x > 0
The concavity of a log function depends upon the parameter, b: • b > 0: increasing function,
concave down
• b < 0: decreasing function, concave up
lim
�
� ( � )=
Endpoint behavior:
lim
� 0
� ( � )=−
increasing function
(asymptotic to vertical axis)
Chapter 1.3 Log Models 4
lim
�
� ( � )=−
Endpoint behavior:
lim
� 0
� ( � )=
decreasing function (asymptotic to vertical axis)
The logarithmic function has no inflection point, that is, it never
changes concavity. An increasing
logarithmic function is concave down.
A decreasing
logarithmic function is concave up.
Chapter 1.3 Log Models 6
Recall exponential functions also have no inflection points, but they are always concave up:
An increasing
exponential function is concave up (dissimilar from logarithmic).
A decreasing
exponential function is concave up (similar to logarithmic).
years (post 1940) life expectancy (yrs)
5 65.2
15 71.1
25 73.1
35 74.7
45 77.4
55 78.8
65 79.3
71 80.2
Example: Women’s life expectancy in US since 1940.
Chapter 1.3 Log Models 8
The scatterplot
suggests a logarithmic model with b > 0 might be appropriate.
Modeling with the graphing calculator: 1. Use STAT, EDIT, Edit to access the
Lists and then enter the values into L1 and L2.
2. Use STAT, CALC, LnReg to execute the natural log regression, saving the regression in an available Yi. Call the model, W(t) and report the model parameters with three decimal digits. 3. Select ZoomStat to plot the
4. According to the model, in what
year did women’s life expectancy in the US reach 75 years old? Round your result to the nearest tenth of a year.
5. According to the model, in what year will women’s life expectancy in the US reach 82 years old?
Round your result to the nearest tenth of a year.
Chapter 1.3 Log Models 12
Using the Solver, women’s life
expectancy in the US reached 75
years old 30.6 years after the end of 1940. This would be in the year
1970.6, that is, 60% of the way through the year 1971.
Using the Solver, women’s life
expectancy in the US will reach 82
years old 106.2 years after the end of 1940. This would be in the year
2046.2, that is, 20% of the way through the year 2047.
This calculation represents an extrapolation using the model.
Chapter 1.3 Log Models 14
Recall the form of the logarithmic function:
�
(
�
)
=
�
+
���
(
�
)
Consider the variable substitution:
�
=
��
(
�
)
�
(
�
)
=
�
+
� �
This last equation is linear. Therefore:
This analysis suggests:
If we first take the natural log of the raw input values, the
relationship between the natural log-transformed input data and the original output data will be linear.
Returning to the women’s life expectancy example:
Chapter 1.3 Log Models 16
Modeling with the TI-84
1. The raw input data is currently in L1. We wish to enter the natural logs of these values in L3. Use STAT, EDIT, Edit to access the
Lists. Scroll to the L3 heading and enter Ln(L1) using 2nd 1 to select L1. At the bottom of the screen you should see “L3 = Ln(L1)”.
Select ENTER. L3 now contains the natural logs of the L1 entries.
L1 L3
years (post 1940) ln-transformed years (post 1940) 5 1.61 15 2.71 25 3.22 35 3.56 45 3.81 55 4.01 65 4.17 71 4.26
Comparing raw input data (years post 1940) to ln-transformed input data (natural log of years post 1940)
Chapter 1.3 Log Models 18
2. We will be creating a scatterplot using Plot2. We need to provide information to the plotting routine. In preparation, select STATPLOT (2nd Y=). Toggle to 2 and ENTER. 3. Turn plot ON.
4. For input (Xlist) select L3 (2nd 3). 5. For output (Ylist) select L2 (2nd 2). 6. QUIT (2nd MODE).
7. Prepare to linearly regress: STAT, CALC, LinReg. Be sure to select L3 for Xlist, that is, the natural log-transformed input values. Select L2 for Ylist, the outputs.
8. Choose a Yi in which to save the regression. Depending upon
calculator model, choose ENTER or Calculate to execute the
regression.
Chapter 1.3 Log Models 20 natural log-transformed
years (post 1940) life expectancy (yrs)
1.61 65.2 2.71 71.1 3.22 73.1 3.56 74.7 3.81 77.4 4.01 78.8 4.17 79.3 4.26 80.2
Example: Women’s life expectancy in US since 1940.
Chapter 1.3 Log Models 22
Observations:
1. Natural Log model:
x = years, aligned to 1940
2. Linear model:
altitude
(000s ft) (inches of Hg)pressure
100 0.33
80 0.82
60 2.14
40 5.56
20 13.76
Problem:
Chapter 1.3 Log Models 24
The scatterplot
suggests a logarithmic model with b < 0 might be appropriate.
Recall, however, that both exponential and logarithmic
functions are concave up when they are decreasing functions. Therefore use the TI-84 to
Chapter 1.3 Log Models 28
When using logarithmic models,
data-alignment may be needed for input values due to following:
1. Log functions require all input values be positive.
2. Because the shape of log functions changes more
radically as input values near 0 (the vertical axis asymptote), alignment may improve the fit.
Reconsider air pressure as a function of altitude.
Alignment options can be tried by adding or subtracting constants from the raw input data.
After experimentation, a
considerably better model can be developed.
Consider a new model by aligning the input data by subtracting 17
Chapter 1.3 Log Models 30
raw
altitude altitudealigned pressure
100 83 0.33
80 63 0.82
60 43 2.14
40 23 5.56
20 3 13.76
Problem:
Use the TI-84 to fit a logarithmic model to the aligned data.
Chapter 1.3 Log Models 32
Comparing model results: 1. Natural Log model:
input: raw altitude data
2. Exponential model: input: raw altitude data 3. Natural Log model:
input: raw altitude - 17
