This curriculum writing project was inspired by the spirit of the NCTM Standards for School Mathematics in the hope that a similar emphasis on applications (connections), communication, and understanding could be incorporated into the standard college curriculum. It is designed to be part of a precalculus course, although it could also fit in Intermediate Algebra. There is probably enough material for two class periods.
Lesson 6.1 Exponentials, 6.2 Properties of Exponential Functions, and 6.3 Logarithms would precede this in the curriculum. I assume that students would have seen the compound interest formula and uses of the formula in modeling inflation. They should have also been exposed to exponential growth and decay.
In the lesson immediately preceding this one, the basic definition of logarithms would have been given (as the inverse of exponentials), and the properties of logarithms would have been deduced. Students should know the difference between common log and natural log, but realize that they have similar properties.
Earlier in the curriculum, students would have been exposed to variation (direct and inverse) and equations of lines through two given points. They should also have some knowledge of linear regression.
Students should know how to use their graphing calculator to
plot functions, use the trace function, plot ordered pairs, and
do linear regression. They should also be able to do
computations involving exponential and logarithmic functions. In
the precalculus course at Seton Hill, a computer algebra system
(Maple V) was used instead of a graphing calculator.
My sources for applications include the following:
Example 1: A Sourcebook of Applications of School Mathematics
(updated to 1990 information - "Critical Thinking" added)
Example 2: A Sourcebook of Applications (graphing added)
Example 3: A Sourcebook of Applications ("Try This" added)
Example 4: General knowledge
Exercise 6 and 7: Mathematics with Applications by Lial & Miller
Exercise 8 and 9: A Sourcebook of Applications.
Exercise 12 and 13: Idea "borrowed" from Algebra 2 by Cohen, et
al. Actual exercises are mine.
Exercises 14 - 17: Equation obtained from an article by Fishman
in Mathematics Teacher, November 1993. Exercises are mine.
Exercise 18: Mathematics with Applications.
6.4 Applications of Logarithms
Why The logarithms studied in the last section are
used in mathematics and applications to:
1. solve exponential equations for variables
appearing in the exponent
2. find a non-linear relationship between variables
3. model phenomena such as sound perception,
earthquake intensity, flicker in a motion picture,
PH level of foods, and city walking speed.
Remember:
y = logb(x) is equivalent to x = by.
logb(xy) = logb(x) + logb(y)
logb(x÷y) = logb(x) - logb(y)
logb(xy) = y logb(x)
Example 1
Ecology In 1990, known world coal reserves were estimated at
8 x 1012 tons, but consumption was running at 5.8 x 109
tons per year. If consumption increases at a rate of
5% per year, how long will the coal last?
If consumption of coal was not increasing, we could
easily calculate the time until known reserves would be
exhausted. How? Using the fact that rate x time is a
measure of total consumption, we see that
8 x 1012 tons = (5.8 x 109 tons/year)(t years).
So t = ______________ years. This is a very long time!
However, due to increasing industrialization in the
developing countries, it is not reasonable to assume
that coal consumption will remain constant. Instead,
ecologists assume that it will increase at a constant
rate each year (in our problem, it increases at 5% per
year).
We saw in Lesson 6.2 that the compound interest formula could be used to model the effects of such things as inflation and the growth of consumption, if a constant percent increase, r, is assumed. In such cases, we saw that n (number of compounding periods per year) = ____. Hence, the formula: r
A = P(1 + )nt
n
becomes: A = P(1 + r)t.
Since P (the consumption in 1990) was 5.8 x 109, and r
is assumed to be 5%, the consumption in year t
(measured since 1990), A, is given by:
A(t) = 5.8(1.05)t billion tons
Graphics Use your graphics calculator to examine the graph of
Calculator A(t) for t from 0 to 60. Use the trace function to
answer the following:
1. What will A be in the year 2000 (t = ___)?
__________ billion tons
2. What about 2025? _______________________ (units?)
3. When will consumption exceed 100 billion tons? ___
Logarithms The last question could have been answered with algebra
and logarithms. For if A = 100, we have:
100
100 = 5.8(1.05)t, so = 1.05t.
5.8
By taking the common log of both sides, we obtain:
100
log ( ) = log(1.05)t = t log(1.05)
5.8
Why?
log(100/5.8)
Hence t = = 58.4 years. Compare to #3!
log(1.05)
So consumption of coal will reach 100 billion 58.4
years after 1990, during the year 2048.
More Logs In order to determine when the known coal reserves (eight trillion tons in 1990) will run out, we need to be able to add the consumption in 1990, 1991, 1992, etc. until the year 1990 + t. In other words, we need to find:
C = A(0) + A(1) + A(2) + ... + A(t)
= P + P(1+r) + P(1+r)2 + ... + P(1+r)t
But a formula for any sum of the form:
S = a + aR + aR2 + ... + aRn (geometric series)
Rn+1 - 1
is given by: S = a
R - 1.
(1+r)t+1 - 1
Using R = 1+r and a = P, we get C = P .
r
1.05t+1 - 1
In our case, C = 5.8 .
.05
To determine when the reserves will be depleted, we
must substitute 8,000 for C (8,000 billion = 8
trillion) and then solve for 1.05t+1:
1.05t+1 = 8000 x .05 ÷ 5.8 + 1 = 69.9655
Now take the log of both sides:
log(1.05t+1) = log(69.9655)
(t+1)log(1.05) = 1.845
So t = 86 years.
Thus, if consumption of coal increases at a rate of 5%
per year, the total known world reserves will be gone
by 2076!
Critical If the known world supply were doubled because of the Thinking discovery of a huge coal deposit, what
would be the effect on the depletion
date?
If the increase in consumption would be held to 4% per
year, what would be the effect on t?
Which change has the greatest effect on t, an increase in supply or a decrease in the growth rate of consumption?
Example 2
Biology An allometry between x and y exists if y = b(xk) for some b and k. Jill, a biologist believes that an allometry exists between baboon face length and skull length. She measures four baboons and obtains:
x face length (mm) 31.0 94.8 131.0 144.25
y skull length (mm) 78.5 108.9 114.7 122.0
What are the "best" estimates of b and k?
Solution >
In many cases, a logarithmic transformation converts a non-linear relationship into one that is linear. Try this with y = b(xk), by taking the log of both sides:
log(y) = log(b) + k log(x) (why?)
Thus if y* = log(y), and x* = log(x), we have:
y* = a0 + a1(x*) where a0 = log(b) and a1 = k.
The relationship between x* and y* is linear!
Calculator Fill-in the remainder of the following table (round to and Graph nearest .001) and plot the pairs on the provided grid:
x* 1.491 2.117
Y* 1.895 2.086
2.0
1.9
1.8
1.7
1.6
1.5
1.4
1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2
Look at the graph that you've completed. Since the
pairs seem to exhibit a linear relationship, it appears
as though the biologist is right. But now, we need to
find b and k. To do this, we'll use two different
methods:
Method 1 - Graphical/Algebraic
1. Using the above graph, attempt to draw a straight
line which "fits" the data as well as possible.
As discussed earlier in the text, one possible
strategy is to draw a line joining the "outside"
points. Your line may differ slightly from others
in your class.
2. Using two convenient points on your line, compute
the slope. This is a1.
3. Use y* = a1x* + a0 and any convenient point
(x*,y*) from your line. Only a0 is unknown, so
you may find it via algebra.
Method 2 - Linear Regression
Use the statistical regression mode of your calculator
to obtain the slope and y-intercept of a line which
fits your data "best" (least squares criterion). You
should obtain a0 = 1.4825 and a1 = .2774 (rounded).
Use your graphics calculator to plot the four (x*,y*)
pairs and the linear regression line found. As in
method 1, the line should closely fit the points.
The Allometry
In either method, we obtained values for a0 and a1,
which gave a line fitting the data. In the following,
we will use the values obtained by method 2:
a0
Since a0 = log(b), b = 10 = 30.3739. k = a1 = .2774
Thus, we have found the allometry for baboon face
length and skull length:
y = 30.3739x.2774.
Example 3
Perception As you know, a movie is a sequence of still pictures
which is shown fast enough to give the impression of
motion to the viewer. If the frequency of the stills
(pictures shown per second) is too small, the movie
flickers; if the frequency is large enough, the brain
perceives smooth motion.
According to research, the minimum frequency, f, at
which the flicker first disappears is proportional to
the logarithm of the light intensity, I, of the screen
image reaching the viewer. Thus:
f = K log(I), K is the proportionality constant
We also know from previous work that I is inversely
proportional to the square of the distance from the
light source (the screen). Hence:
I = k/d2, so f = K log(k/d2) = K[log(k) - 2log(d)]
why?
The Problem>
Suppose that Alicia notices a flicker from the film
(she is sitting too close) and doubles her distance
from the screen. What effect will this have on f?
The Solution>
Let f1 be the minimum flicker-free frequency (a tongue
twister!) in the original seat at distance d, and let
f2 be the frequency at distance 2d. Then:
f1 = K[log(k) - 2log(d)] and
f2 = K[log(k) - 2log(2d)] = K[log(k) - 2log2 - 2log(d)]
Why?
So f1 - f2 = Klog(4) = 0.602K.
Hence, by doubling the distance from the screen, f is
only reduced by 0.602K. This may not be enough to
eliminate the flicker in the film!
Try This What if Alicia triples her distance from the screen.
What is the effect on f?
Critical Suppose Karl cannot change seats (the movie theater is
Thinking full), but he perceives a flicker. How can he solve
the problem? Hint: consider f = K log(I).
Example 4
Chemistry The pH factor determines whether a substance is
alkaline, neutral, or acidic depending if pH is greater
than, equal to, or less than 7 respectively. If H+ is
the hydrogen ion concentration in moles, pH is
determined by: pH = -log(H+).
1. Tomato juice has H+ = 6.3 x 10-5 moles. What is
the pH value?
pH = -log(6.3 x 10-5) = 4.2
So tomato juice is acidic.
2. Milk of magnesia has H+ = 3.2 x 10-11.
pH is ______ and milk of magnesia is _____________
Critical What is the H+ value for a neutral substance?
Thinking
Exercises & Problems
Communicate
1. Explain why the natural logarithm could have been used in
example 1 and 2, when we took the logarithm of both sides of
an exponential equation.
2. In example 3, common logs were used to model the frequency
at which movie flicker disappears. Could natural logs be
used throughout? Explain!
3. Give the product, quotient, and power rules for logarithms in your own words. E.g. "The log of a product is ..."
Practice and Apply
4. If y = a(bx), what transformation would you use to create a
linear relationship between x* and y*?
Business A famous quote by Ben Franklin says "A penny saved
is a penny earned." But suppose that you deposit the penny in a
savings account paying 5% interest, compounded monthly.
5. How long would it take until the amount is $1,000? Until it
is $1 million? Assume that the bank does not round interest
accumulations until the money is finally withdrawn.
Business The Rule of 72 states that it will take
approximately 72 ÷ r years for a deposit to double, if interest
is compounded annually at r%.
6. Using the compound interest formula and logarithms, find how
long it would take until your $1 deposit doubles if the
rates are 4%, 6%, and 8% (round to the nearest year).
7. Compare the answers obtained in #6 with those given by the
Rule of 72.
Biology Michaela believes that there is an allometry between the height, x, and the weight, y, of humans. A study reveals the average weights for boys and girls of heights 60 and 70 inches are: Weight
Height Boys Girls
60 104 94
70 157 143
8. Find the allometry relationship for boys (use log
transformations and the point-slope formula).
9. Find the allometry relationship for girls.
Astronomy Let d be the average distance of a planet from the sun, measured in astronomical units (A.U.). One A.U. is the average distance of Earth from the sun, approximately 93 million miles. Let T be the period of the planet, defined to be the length of time in years required to make one orbit of the sun. The following table gives d and T for the nine planets:
Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune Pluto
d 0.387 0.723 1.000 1.523 5.203 9.541 19.190 30.086 39.507
T 0.241 0.615 1.000 1.881 11.861 29.457 84.008 164.784 248.350
10. Find a relationship between log(T) and log(d).
11. Use the result of #10 to find the relationship that Kepler
found in 1620, giving T as a function of d.
Sociology It has been observed that walking speed, s, measured in miles per hour is related to the population of a city, by:
s = 1.38 log(p) + .03
12. Find the walking speed in Pittsburgh (population 370,000),
New York (7.3 million), and Greensburg, PA (16,000).
13. What is the population of a town, if the walking speed is
5.55 mph?
Ecology An analysis of the yearly consumption, A, of natural gas in the United States between 1900 and 1970 showed that:
A(t) = 0.3(1.063)t trillion cubic feet (t.c.f.), for 0 t 70
14. Use your graphics calculator to graph A(t). Use the trace
function to determine when A reached 3 t.c.f.
15. Use logarithms to determine when A reached 20 t.c.f.
16. Find the total consumption between 1900 and 1970.
17. Data shows that United States consumption of natural gas
ceased its exponential growth after 1970. In fact, in 1990
A was 18.55 t.c.f. What would the consumption have been in
1990, if the same growth had continued after 1970?
Archeology As discussed in Lesson 6.2, carbon-14 (C-14)
decay is modeled by an exponential function:
y = y0e-.000123t, where y is the amount of C-14 remaining
t years after a living organism's death.
18. A round table hanging in Winchester Castle (Great Britain) was supposed to be the famous table used by King Arthur of the 5th century. Chemical analysis in the 1950's showed that the table had 91% of the amount of C-14 present in living wood (.91y0). Find the approximate age of the table. Was it King Arthur's round table?
Text Answers:
Page 1 t = 1,379 years
Page 2 n (number of compounding periods per year) = 1
1. (t = 10) 9.5 billion tons
2. 32.2 billion tons
3. 2048
Why?: power rule for logarithms
Compare to #3 -- 58.4 gives 2048, which is the same
year as found in #3
Page 3 Critical Thinking
1. If deposits doubled, C = 16,000 which gives t=100, or 14 more years of consumption
2. If r is 4% (and C = 8,000), t=107, so 21 more years.
3. The decrease in growth rate has the greatest
effect -- C changed by 100%, yielding a change of
14 years in t, but r changed by only 20%, and
yielded a 21 year change
Page 4 Why?: log(bxt) = log(b) + log(xt) by product rule
= log(b) + t log(x) by power rule
Table: 1.491 1.977 2.117 2.159
1.895 2.037 2.060 2.086
Page 6 Why?: K log(k/d2) = K[log(k) - log(d2)] quotient rule
= K[log(k) - 2log(d)] power rule
Why?: -2log(2d) = -2[log(2) + log(d)] product rule
Page 7 Try: log(3d) = log(3) + log(d), so f1 - f2 = K(2log(3))
= K log(9) = .954K
Critical: reduce I by wearing sunglasses or squinting
Milk of magnesia: pH is 10.5 and it is alkaline.
Critical: pH = 7, so -7 = log(H+), so H+ = 10-7 moles.
Answers to Selected Exercises
1. Answers vary, but the main idea is that the properties of
logs used apply for logs of any base.
2. Natural log could have been used since ln(I) = log(I)/log(e)
and so K ln(I) = (1/log(e))K log(I). Just use a different
constant of variation.
3. The log of a product is the sum of the logs of the factors.
The log of a quotient is the difference of the logs.
The log of an exponential is the exponent multiplied by the
log of the base.
4. Since log(y) = log(a) + x(log(b), use y* = log(y) and x* = x
Then y* = a0 + a1x* is linear, with a0 = log(a) and
a1 = log(b).
5. A = .01(1 + .05/12)n.
a. If A=1000, log(1000/.01) = n log(1.00417)
n = 2766.6 months = 230.6 years.
b. If A=1,000,000 a similar calculation gives 4426.6
months or 368.9 years.
NOTE: The assumption is required -- otherwise the bank would
constantly pay 0.00 interest on the original penny, and
hence the money would never grow.
6. 2 = 1(1 + r)n. So log(2) = n log(1 + r),
and n = log(2) ÷ log(1 +r).
If r = 4%, n = 17.7 years = 18 years (rounded).
If r = 6%, n = 11.9 years = 12 years.
if r = 8%, n = 9.01 years = 9 years.
7. Rule of 72 gives 72/4 = 18, 72/6 = 12, and 72/8 = 9 years.
10. log(T) = 1.5 log(d)
11. log(T) = log(d1.5), so T = d1.5 or T2 = d3.
12. Pittsburgh: 1.38 log(370,000) + .03 = 7.71 mph.
Similarly, NYC: 9.50 and Greensburg: 5.83 mph.
13. 5.55 = 1.38 log(p) + .03, so log(p) = 4, so p = 104.
14. Approximately 1937 (t=37.9)
15. 20 = .3(1.063)t, so log(20/.3) = t log(1.063).
Thus t = 68.7, so 1968.
18. .91y0 = y0e-.000123t. Thus ln(.91) = ln(e-.000123t).
So -.000123t = ln(.91) and t = 767 years old. It cannot be King Arthur's Table, since the tree was cut down around the year 1200.