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Supplementary Lesson:
Log-log and Semilog
Graph Paper
Chapter 7 looks at some elementary functions of algebra, including linear, quadratic,
power, exponential, and logarithmic. The following supplementary section on log-log
and semilog graph paper is intended for use at the conclusion of this chapter. This
section provides teaching materials, explorations, teacher notes, and solutions. You
can use this lesson in your regular teaching plan or assign it for individual projects,
group investigations, or extra credit.
Foerster_IRB2_023-025 9/24/03 8:27 PM Page 25
Instructor’s Commentary
Objective
Given an exponential, power, or logarithmic
function, plot its graph on semilog or log-log graph
paper.
Class Time
2 days
The listing of different kinds of graph paper should
help the students keep track of which kind to use
for a particular problem.
Sometimes students may get an error message
when they are doing a regression on their grapher.
When the grapher does exponential regression, it
re-expresses the data using (x , log y), which is why
the grapher is unable to do exponential regression
on a data set that includes 0 as one of the y-values.
Likewise, power regression re-expresses the data
using (log x, log y), so x and y must be positive
numbers.
Homework Assignment
Day 1: Q1–Q10, Problems 1, 3, 5, 7
Exploration Notes
Day 2: Problems 9, 11, 13, 14
In Mathematical Models and Log Graphs, students
will plot points on semilog and log-log graph paper
and make connections to the add–multiply (exponential) and multiply–multiply (power) properties.
Allow about 20 minutes to complete.
Important Terms and Concepts
Semilog graph paper
Log-log graph paper
Lesson Notes
Exponential functions can be very hard to read for
points with large y-values and for x-values near the
horizontal asymptote. One way of showing the
function more clearly is to change the y-scale to
log y. This makes the y-values smaller for large
y-values and negative for positive values of y near
zero. Students may have seen such graphs before.
For instance, graphs displaying financial information
over a period of time, such as the growth of a
mutual fund, often are semilog graphs.
In the lesson we see that the exponential function
y = 1000(0.65x ) is linear when graphed on semilog
paper. Therefore, it follows that if the points plotted
on the exponential function were re-expressed as
(x, log y), then these re-expressed points would be
linear on arithmetic graph paper. Similarly, power
functions are linear on log-log paper, and a reexpression of the power data as (log x, log y) would
be linear on arithmetic graph paper.
It may not be obvious which property a set of data
exhibits, so Log-log and Semilog Graph Paper gives
students practice in finding which regression best
fits the data. The results are then verified by using
the appropriate graph paper to show the linear
relationship. This exploration should take about
25 minutes to complete.
Problem Notes
• Problems 1–8 reinforce the skills and concepts in
the examples and explorations.
• Problems 9–14 demonstrate that the log y versus x
is linear for exponential functions and that the
log y versus log x is linear for power functions.
In Example 1, the data are plotted on log-log paper
to demonstrate that the relationship between the
variables is a power function. In addition, be sure
students note the multiply–multiply relation in the
x and y variables. Students will revisit parametric
equations and see how to use them to graph the
log–log relationship.
Example 2 demonstrates how to use semilog paper
to graph a multiply–add relationship, which models
a logarithmic function.
Precalculus with Trigonometry: Instructor’s Resource Book, Volume 2
©2004 Key Curriculum Press
Log-log and Semilog Graph Paper Instructor’s Commentary / 25
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Log-log and Semilog Graph Paper
Student Lesson
In many real-world situations, the y-values of a function span a wide range. For
instance, suppose the number of bacteria in a culture at the end of any one
hour has dropped to 65% of what it was at the beginning of the hour. If there
were initially 1000 million bacteria present, then the number of millions of
bacteria, y, is given by the exponential function
y = 1000 • 0.65x
where x is time in hours.
y
y
1000
1000
100
500
10
x
0
0
5
10
15
x
1
5
10
15
0
Figure A
Figure B
Figure A shows the graph of this function for the first 15 hours. It is hard to
read the graph for x between 10 and 15, even though the axes have a large
vertical dimension. In Figure B the vertical scale has been compressed for
larger y-values and expanded for smaller y-values. The spaces represent the
logarithms of the y-values. You may already have seen such semilog graph
paper in the explorations. On this graph it is just as easy to read y for larger
x-values. For exponential functions, this graph paper has the added advantage
that the points lie in a straight line! The graph of a power function turns out to
be a straight line on log-log paper, on which the spaces in the x-direction also
represent logarithms, of the x-values.
OBJECTIVE
Given an exponential, power, or logarithmic function, plot its graph on
semilog or log-log graph paper.
26 / Log-log and Semilog Graph Paper Student Lesson
Precalculus with Trigonometry: Instructor’s Resource Book, Volume 2
©2004 Key Curriculum Press
Foerster_IRB2_026-031 9/24/03 8:28 PM Page 27
Straight-line Semilog and Log-log Graphs
To see why an exponential function has a straight-line graph on semilog paper,
it is sufficient to show that log y varies linearly with x. To do this, take the log
of both sides of the exponential function equation.
y = 1000 • 0.65x
log y = log (1000 • 0.65x )
Write “log” in front of both sides.
log y = log 1000 + log 0.65x
The log of a product equals the sum of
the logs.
log y = log 1000 + x • log 0.65
The log of a power equals the exponent
times the log of the base.
log y = 3 D 0.1870…x
By calculator
N log y varies linearly with x, Q.E.D.
Form: y = ax + b
The graph in Figure B is a straight line with a negative slope. In Problem 13 of
this lesson you’ll prove this property in general for exponential functions. In
that problem you’ll also prove that the graph of a power function is a straight
line on log-log graph paper.
U EXAMPLE 1
Snake Skin Problem: Snakes shed their skins periodically. The area of the skin
depends on the length of the snake. Suppose you measured the areas in this
table.
Length (cm)
Area (cm2)
5
10
20
60
100
2.25
9
36
324
900
a. Plot the data on log-log graph paper. What do you notice about the points?
b. Show by regression that a
power function fits these data.
Write its particular equation.
y
1000
c. Predict the skin area of an
anaconda snake 7 meters long.
Surprising?
d. Use your equation from part b
to plot on your grapher the log
of area as a function of log of
length. What do you notice
about the graph?
e. Prove algebraically that log of
area is a linear function of log
of length. What do you notice
about the slope of this graph?
Solution
a. Let x = length in cm and let
y = area in cm2. Figure C shows
the graph of area as a function
of length on log-log paper. The
points lie in a straight line.
Precalculus with Trigonometry: Instructor’s Resource Book, Volume 2
©2004 Key Curriculum Press
100
10
x
1
10
100
1
Figure C
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b. By power regression, r = 1. So a power function fits the data. The particular
equation is y = 0.09x 2.
c. 7 meters is 700 cm, so y = 0.09(7002) = 44,100 cm2. Surprising!
d. Put your grapher in parametric mode. Then enter the equations like this:
x = log t
y = log (0.09 • t 2)
The graph is shown in Figure D. It is a straight line.
3
log y
2
1
1
log x
2
Figure D
e.
log y = log (0.09x 2)
Take the log of both sides.
log y = log 0.09 + 2 log x
log of a product and log of
a power
log y = D1.0457… + 2 log x
By calculator
N log y is a linear function of log x, Q.E.D.
Form: log y = a(log x ) + b
The slope of the linear graph equals the exponent of x. You can confirm
this fact geometrically by picking two points on the graph, measuring the
run and the rise with a ruler, and observing that rise
run = 2. Figure E shows
how you can do this.
y
3
4
5
1000
2
100
1
Ruler
10
1
2
3
4
5
x
1
10
100
1
Figure E
28 / Log-log and Semilog Graph Paper Student Lesson
V
Precalculus with Trigonometry: Instructor’s Resource Book, Volume 2
©2004 Key Curriculum Press
Foerster_IRB2_026-031 9/24/03 8:28 PM Page 29
U EXAMPLE 2
a. By regression analysis, show numerically that a logarithmic function fits
these x- and y-values better than a linear, exponential, or power function.
x
y
2
10
50
300
900
2.4
5.0
7.6
10.4
12.2
b. Write the particular equation of the best-fitting logarithmic function.
c. Plot the points on multiply-add semilog graph paper. Show that the points
lie on a line.
d. Calculate y if x is 20. Show that this point lies on the graph in part c.
e. Calculate x if y is 15.
Solution
a.
Linear: r = 0.8331…
Logarithmic: r = 0.999978… Best fit!
Exponential: r = 0.7133…
Power: r = 0.9736…
b. Equation is y = 1.3067… + 1.6001… ln x. Paste it into the y= menu (without
rounding).
c. Figure F shows that the points lie in a straight line. You can get multiplyadd semilog graph paper by rotating regular add-multiply paper 90−
clockwise.
y
15
Regression line
fits closely.
10
5
(20, 6.1...)
is on the line.
x
1
10
100
1000
1
Figure F
d. If x = 20, then y = 1.3067… + 1.6001… ln 20 = 6.1005…
The point (20, 6.1005…) is on the regression line in Figure F.
e. If y = 15, then 15 = 1.3067… + 1.6001… ln x.
x ≈ 5204.2…
Precalculus with Trigonometry: Instructor’s Resource Book, Volume 2
©2004 Key Curriculum Press
Use the solver to find x numerically.
V
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DEFINITIONS: Kinds of Graph Paper
Arithmetic graph paper (“ordinary” graph paper) has linear scales on
both axes.
Log-log graph paper has logarithmic scales on both axes.
Add-multiply semilog paper has a logarithmic scale on the vertical axis only.
Multiply-add semilog paper has a logarithmic scale on the horizontal
axis only.
Problem Set
Do These Quickly
5 min
Q1. log 5 + log 7 = log –?–
Q2. log 18 D log 3 = log –?–
Q3. 2 log 7 = log –?–
Q4. log10 10 = –?–
Q5. log5 5 = –?–
Q6. Logarithmic functions have the —?— D —?—
pattern.
Q7. What pattern do the y-values of a quadratic
function follow for regularly spaced x-values?
Q8. Name the kind of sequence: 23, 30, 37, 44, …
Q9. x 60 ÷ x 10 = x–?–
Q10. Find the area of a right triangle with one leg
3 cm and hypotenuse 5 cm.
For the exponential functions in Problems 1 and 2,
a. Calculate the y-values for the given values
of x.
b. Plot the points on add-multiply semilog
graph paper.
c. Show that the points lie on a straight line.
1. y = 1.5 • 2x ; x = 1, 3, 5, 7, 9
2. y = 800 • 0.6x ; x = 2, 4, 6, 8, 10
For the power functions in Problems 3 and 4,
a. Calculate the y-values for the given values
of x.
b. Plot the points on log-log graph paper.
c. Show that the points lie on a straight line.
d. Measure the slope with a ruler and show
that it equals the exponent in the equation.
30 / Log-log and Semilog Graph Paper Student Lesson
3. y = 700xD1.3; x = 1, 5, 10, 30, 100
4. y = 5x 0.8; x = 1, 6, 10, 40, 100
For the logarithmic functions in Problems 5 and 6,
a. Calculate the y-values for the given values
of x.
b. Plot the points on multiply-add semilog
graph paper.
c. Show that the points lie on a straight line.
5. y = 2 + 3 ln x; x = 1, 4, 10, 200, 1000
6. y = D1 + 2 log x; x = 3, 8, 20, 100, 500
7. Show that an exponential function graph is not
straight on log-log paper by plotting the data
from Problem 1 on log-log paper.
8. Show that a power function graph is not
straight on semilog paper by plotting the data
from Problem 4 on semilog paper.
For the exponential functions in Problems 9 and 10,
plot on your grapher log y as a function of x. Use
suitable x- and y-windows. Sketch the result.
9. y = 5 • 3x
10. y = 20 • 0.8x
For the power functions in Problems 11 and 12, plot
on your grapher log y as a function of log x. You
may use parametric mode as in Example 1. Use
suitable x- and y-windows. Sketch the result.
11. y = 90xD2
12. y = 2x 3
13. Proof Problems:
a. Prove that for the exponential function
y = 5 • 3x, log y is a linear function of x.
b. Prove in general that for the exponential
function y = ab x, log y is a linear function
of x.
Precalculus with Trigonometry: Instructor’s Resource Book, Volume 2
©2004 Key Curriculum Press
Foerster_IRB2_026-031 9/24/03 8:28 PM Page 31
c. Prove that for the power function y = 2x 3,
log y is a linear function of log x.
d. Prove in general that for the power function
y = ax b, log y is a linear function of log x.
e. How can you conclude that in a logarithmic
function, y is a linear function of log x?
14. Height-Weight Historical Problem: Before there
were calculators to do regression analysis
efficiently, log-log and semilog graph paper
were used to help determine what kind of
function fits a given set of data. The kind of
paper that gave the straight-line graph
indicated the kind of function to use. Suppose
that these average weights have been recorded
for humans.
Height (in.)
Weight (lb)
10
20
30
40
50
60
1.7
8.5
21.5
41.6
69.5
105.7
Precalculus with Trigonometry: Instructor’s Resource Book, Volume 2
©2004 Key Curriculum Press
a. Plot the data on add-multiply semilog paper.
Plot it again on log-log paper. Which graph
seems to be more nearly a straight line?
b. Based on your answers to part a, what kind
of function fits the data more closely,
exponential or power?
c. Find algebraically the particular equation of
the function in part b. Use the first and the
last data points to find the constants in the
equation. (This was the way particular
equations were found in the days before
calculators had regression analysis built in.)
d. Do exponential regression and power
regression on the given data. Does the
regression analysis confirm your conclusion
and equation? How can you tell?
e. Predict the weight of a 90-in.-tall giant using
your equation from part c and again using
the regression equation in part d. How
closely do the two answers agree?
Log-log and Semilog Graph Paper Student Lesson / 31
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Name:
Group Members:
Exploration: Log-log and Semilog Graph Paper
Date:
Objective: Given a table of data with irregularly spaced x-values, find by regression the
particular equation of the best-fitting function and show that the graph is a straight line on
the appropriate kind of graph paper.
For Problems 1–3, use this data.
x
f (x)
3
4.1
For Problems 4–6, use this data.
x
g(x)
2
261
7
24
5
87
8
37
9
43
11
136
20
16.5
12
212
70
3.7
100
2.4
1. By regression, find the particular equation of the
best-fitting linear, exponential, or power function.
2. Plot the data on this semilog graph paper. What do
you notice about the points?
f (x)
4. By regression, find the particular equation of the
best-fitting linear, exponential, or power function.
5. Plot the data on this log-log graph paper. What do
you notice about the points?
1000
g(x)
1000
100
100
10
10
x
1
5
10
15
0
3. Calculate f (0) and f (15). Plot these points on the
graph. Do these points lie on the same straight line
as the given data points?
32 / Log-log and Semilog Graph Paper Explorations
x
1
10
100
1
6. Calculate g(1) and g(50). Plot these points on the
graph. Do these points lie on the same straight line
as the given data points?
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©2004 Key Curriculum Press
Foerster_IRB2_032-040 9/24/03 8:28 PM Page 33
Name:
Group Members:
Exploration: Log-log and Semilog
Graph Paper continued
For Problems 7–9, use this data.
Date:
10. What kind of graph paper gives a straight-line
graph for
x
h(x)
1
5.3
6
11.8
Exponential functions?
10
17.0
Power functions?
13
20.9
18
27.4
Linear functions?
7. By regression, find the particular equation of the
best-fitting linear, exponential, or power function.
11. Let y = 13 • 7x. What kind of function is this? How do
you tell?
8. Plot the data on this arithmetic graph paper.
What do you notice about the points?
12. Take the log of both sides of the equation in
Problem 11. By appropriate use of the properties of
logs, show that log y varies linearly with x.
h(x)
30
13. From the semilog paper in Problem 2, measure to
the nearest 0.1 mm the following distances on the
vertical scale:
25
y-values
Distance, mm
Ratio
log 2 (3, 4, . . .)
1 to 2
1 to 3
20
1 to 4
1 to 5
15
1 to 6
1 to 7
1 to 8
10
1 to 9
1 to 10
14. Make another column in the table above and record
5
Distance from 1 to 2 (3, 4, …)
Distance from 1 to 10
x
5
10
15
20
9. Calculate h(0) and h(20). Plot these points on the
graph. Plot a straight line through these points. How
does the given data relate to the line?
Round the ratios to 2 decimal places.
15. Make another column in the table above and record
log 2 (3, 4, …). That is, record log 5 in the 1 to 5 row
and so on. Round to 2 decimal places.
16. Based on the table above, what do the distances on
the y-scale of semilog paper represent?
17. What did you learn as a result of doing this
Exploration that you did not know before?
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Name:
Group Members:
Exploration: Mathematical Models and Log Graphs
Date:
Objective: Plot graphs of exponential or power functions on semilog or log-log graph
paper, respectively.
Coffee Cup Problem: You pour a cup of coffee. Three
minutes after you pour it, you find that it is 52.7−F above
room temperature. You record its temperature each
2 minutes thereafter, finding the following values:
x (min)
f (x) (−−F)
3
52.7
5
28.8
7
15.7
9
8.6
11
4.7
3. Exponential functions have the add-multiply
property. By exponential regression find the
particular equation of the best-fitting exponential
function. Write the correlation coefficient.
4. Use your equation to find f (0). Show that the answer
agrees with the pattern of the points in Problem 1.
If the room temperature was 70−, how hot was the
coffee when it was poured?
1. Plot these points on this semilog graph paper. The
scale on the y-axis is logarithmic, with the distance
from the x-axis representing the logarithm of the
y-value. For y between 1 and 10, each space
represents 1 unit. For y between 10 and 100, each
space represents 10 units, and so forth. Connect
the points. If any point does not lie on a straight
line, go back and check your work.
5. Put log f (x) in a third list in your grapher. Then do
linear regression with x and log f (x). Write the result
here. Write the correlation coefficient.
f (x)
1000
6. On your grapher, plot the linear function from
Problem 5 along with the points from the lists of
x and log f (x). Use an x-window of [0, 15] and a
y-window of [0, 3]. Sketch the result here. What
relationship do you notice between this graph and
the graph in Problem 1?
100
10
x
1
5
10
15
0
2. Show in the table that the points have, approximately,
the add-multiply property. State the result verbally.
“Adding 2 to x multiplies f (x) by about —?—.”
34 / Log-log and Semilog Graph Paper Explorations
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Name:
Group Members:
Exploration: Mathematical Models and
Log Graphs continued
Shark Problem: From great white sharks caught in the
past, fishermen find the following weights and lengths.
x (ft)
g(x) (lb)
5
10
15
20
75
600
2025
4800
7. Plot these values on this log-log graph paper. The
vertical and horizontal scales are both logarithmic.
Connect the points. If the points do not lie in a
straight line, find your mistake.
Date:
10. Calculate the weight of an 8-ft-long shark. Plot the
point on the graph paper in Problem 7, thus showing
that the point is on the line.
11. From fossilized sharks’ teeth, naturalists think there
were once great white sharks 100 ft long. Based on
your mathematical model, how heavy would such a
shark be? Surprising?
g (x)
10000
12. Put two more lists in your grapher for the shark data
of Problem 7, one for log x and one for log g(x). Do
linear regression for log g(x) as a function of log x
and write the equation. What evidence do you have
that a linear function fits these transformed values
exactly?
1000
100
x
10
10
100
13. On your grapher, plot the linear function along with
the points from the lists of log x and log g(x). Use an
x-window of [0, 2] and a y-window of [1, 4]. Sketch
the result. How does the graph compare with the
log-log graph in Problem 7?
1
8. Show in the table that the points have the multiplymultiply property. State the result verbally.
“Multiplying x by —?— multiplies g(x) by —?—.”
14. What did you learn as a result of doing this
Exploration that you did not know before?
9. Power functions have the multiply-multiply property.
Do power regression on the given points. Write the
particular equation. What evidence do you have that
the fit is exact?
Precalculus with Trigonometry: Instructor’s Resource Book, Volume 2
©2004 Key Curriculum Press
Log-log and Semilog Graph Paper Explorations / 35
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Solutions
Problem Set
b., c.
y
Q1. log 35
Q2. log 6
Q3. log 49
Q4. 1
Q5. 1
Q6. multiply-add
1000
Q7. Constant 2nd differences
Q8. Arithmetic
Q10. 6 cm
100
Q9. x
50
2
1. a. x
y
10
1
3
3
12
5
48
7
192
9
768
x
1
5
10
15
0
b., c.
3. a. x
y
y
1
1000
100
700
5
86.3847…
10
35.0831…
30
8.4108…
100
1.7583…
b., c.
y
1000
10
100
x
1
5
10
15
0
2. a. x
y
2
288
4
103.68
6
37.3248
8
13.436928
10
4.83729408
10
x
1
1
10
100
1000
d. Slope = D1.3
36 / Log-log and Semilog Graph Paper Solutions
Precalculus with Trigonometry: Instructor’s Resource Book, Volume 2
©2004 Key Curriculum Press
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4. a. x
y
1
b., c.
5
6
y
20.9648…
10
31.5478…
40
95.6352...
100
199.0535…
10
b., c.
y
1000
x
0
1
100
7.
10
100
10
100
1000
y
1000
10
100
x
1
10
1
100
1000
10
d. Slope = 0.8
5. a.
x
y
1
2
4
6.1588…
10
8.9077…
200
17.8949…
1000
22.7232…
x
1
1
8.
1000
y
1000
b., c.
y
30
100
20
10
10
x
0
10
1
6. a.
x
100
1000
y
3
D0.0457…
8
0.8061…
20
1.6020…
100
3
500
4.3979…
Precalculus with Trigonometry: Instructor’s Resource Book, Volume 2
©2004 Key Curriculum Press
x
1
0
25
50
75
100
Log-log and Semilog Graph Paper Solutions / 37
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9.
Log-log:
log y
y
1000
x
1
1
100
10.
log y
x
1
10
1
11.
log y
x
1
1
10
100
1000
Log-log gives more nearly a straight line.
log x
1
b. Power.
1
c. a • 10b = 1.7, a • 60b = 105.7
Dividing, 6b =
12.
log y
a=
1
e. 14c: 0.0084…(90)2.3049… = 269.1302… lb.
14d: 0.0084…(90)2.3033… = 269.5358… lb.
13. a. y = 5 • 3x ⇒ log y = log (5 • 3x) = log 5 + (log 3)x
b. y = ab x ⇒ log y = log (ab x) = log a + (log b)x
Log-log and Semilog
Graph Paper Explorations
c. y = 2x 3 ⇒ log y = log (2x 3) = log 2 + 3 log x
d. y = ab x ⇒ log y = log (ax b) = log a + b log x
e. y = a + c logb x = a + c •
1.7
1.7
=
= 0.0084…; y = 0.0084…x2.3049…
10b 102.3049…
d. y = 1.3499…(1.0820…)x has r = 0.9670…
y = 0.0084…x2.3033… has r = 0.9999…
Power regression equation is very close to the equation in
part b and has a very good r-value.
log x
1
105.7
105.7
⇒ b = log6
= 2.3049…;
1.7
1.7
log x
c
=a+
• log x
log b
log b
Log-log and Semilog Graph Paper Problem
1. LinReg: f (x) = 22.22220…x D 99.6007…, r = 0.8956…
ExpReg: f (x) = 1.1099… • 1.5492…x, r = 0.9999…
PwrReg: f (x) = 0.1561…x2.7734…, r = 0.9799…
The exponential function fits the best.
14. a. Add-multiply semilog:
y
1000
100
10
x
1
0
5
38 / Log-log and Semilog Graph Paper Solutions
10
Precalculus with Trigonometry: Instructor’s Resource Book, Volume 2
©2004 Key Curriculum Press
Foerster_IRB2_032-040 9/24/03 8:28 PM Page 39
7. LinReg: h(x) = 1.3x + 4, r = 1
ExpReg: h(x) = 5.7733… • 1.0995…x, r = 0.9697…
PwrReg: h(x) = 5.0032… • x 0.5517…, r = 0.9894…
The linear function fits the best.
2. Points are in a line.
f (x)
1000
8. The points are in a line.
h(x)
30
100
25
20
10
15
x
1
5
10
15
10
0
3. f (0) = 1.1099…; f (15) = 788.7983…
Points lie on the line.
4. LinReg: g(x) = D1.4425…x + 118.4618…, r = D0.5931…
ExpReg: g(x) = 93.6080… • 0.9597…x, r = D0.9194…
PwrReg: g(x) = 598.1917…xD1.1978…, r = D0.9999…
The power function fits the best.
5
x
5. The points are in a line.
5
g(x)
10
15
20
9. h(0)=4, h(50)=69
The points lie on the line.
1000
10. Linear: Arithmetic graph paper
Exponential: Add-multiply semilog graph paper
Power: Log-log graph paper
11. Exponential function (the x is in the exponent).
12. log y = log(13 R 7x) = log 13 + x log 7
100
13., 14., 15.
10
x
1
10
100
1
6. g(1) = 598.1917…, g(50) = 5.5163…
Points lie on the line.
y-values
Distance
Ratio
log
1 to 2
11.4
0.30
0.30
1 to 3
18.1
0.48
0.48
1 to 4
22.8
0.60
0.60
1 to 5
26.5
0.70
0.70
1 to 6
29.5
0.78
0.78
1 to 7
32.5
0.86
0.85
1 to 8
34.2
0.90
0.90
1 to 9
36.2
0.96
0.95
1 to 10
37.9
1.00
1.00
16. The distances on the vertical axis represent the common
logarithm of the function’s value.
17. Answers will vary.
Precalculus with Trigonometry: Instructor’s Resource Book, Volume 2
©2004 Key Curriculum Press
Log-log and Semilog Graph Paper Solutions / 39
Foerster_IRB2_032-040 9/24/03 8:28 PM Page 40
Mathematical Models and Log Graphs Problem
1.
7.
g(x)
10000
f (x)
1000
1000
100
10
100
x
1
10
5
0
15
x
10
10
1
2.
+2
+2
+2
+2
x
f (x)
3
52.7
5
28.8
7
15.7
9
8.6
11
8.
x (ft)
× 0.546
×2
× 0.546
× 0.546
×2
× 0.546
4.7
Adding 2 to x multiplies f (x) by 0.546.
3. f (x) = 130.3922… • 0.7392…x, r = 0.9999…
4. f (0) = 130.3922…
(See square point on graph in Problem 1.)
The coffee was about 130.4− + 70−, or about 200.4−.
5. y = 2.1152… D 0.1312x, r = 0.9999…
100
g(x)
5
75
10
600
15
2025
20
4800
×8
×8
9. g(x) = 0.6x3, r = 1
The power function fits exactly because the correlation
coefficient is exactly 1.
10. g(8) = 0.6(8)3 = 307.2
(See square point on graph in Problem 7.)
11. g(100) = 0.6 • 1003 = 600000 lbs = 3000 tons
3
12. g(x) = 3x + log 0.6, r = 1
The correlation coefficient is exactly 1, so the linear function
fits exactly.
2
13.
6.
log f(x)
log f (x)
4
1
x
5
10
15
3
2
x
1
2
14. Answers will vary.
40 / Log-log and Semilog Graph Paper Solutions
Precalculus with Trigonometry: Instructor’s Resource Book, Volume 2
©2004 Key Curriculum Press