1
Exponential and Logarithmic Functions
Importance in Environmental Studies
Logarithms enable infinitely large and infinitesimally small numbers to register
in our minds. “The utility of the logarithm is aptly demonstrated by its ability to
represent nume
rical excesses in comprehensible terms.” It appears that human
senses perceive the world though a logarithmic lens.
2
Algebraic Mathematical Operations of Logarithms
DIRECT OPERATIONS
INVERSE OPERATIONS
a) addition
subtraction
a + b =
c
b = c
–
a
b) multiplication
division
ab = c
b = c/a
c) exponent or power
logarithm
b
a
= c
a = log
b
c
Logarithm defined:
The logarithm answers the question:
To what exponent (x), must you raise the base (b) to get the number
(y)?
for base 10 log....How many orders of magnitude (powers of 10)?
means
3
http://micro.magnet.fsu.edu/primer/java/scienceopticsu/powersof
10/
Note: log
10
x = 0.434 ln(x)
The logarithm answers the question:
To what exponent (x), must you raise the base (b) to get the number (y)?
for base 10
log....How many orders of magnitude (powers of 10)?
means
4
Logarithmic rules of thumb:
For a number written in standard scientific notation, the exponent
of the ten is roughly its logarithm.
The log
10
of a number tells you the order of magnitude (i.e. how
many powers of ten
)
Arguments in exponents and logarithms must be dimensionless.
**Logs allow you to solve for exponents. Exponents “undo” logs.**
5
Common log scales in Environmental Science
pH scale
Richter scale
Electromagnetic
Decibel scale
+Many more
What is the concentration of hydrogen ions [H+] in Ammonia?
pH = 12
12 =

log
10
[H+]
[H+]= 10

12
How do you convert (back) from Ammonia’s concentration of hydrogen ions [H+] to
pH?
Take the log of both sides.
pH =

log
10
[H+]
6
What is the concentration of hydrogen ion
s [H+] in Vinegar?
pH = 3
3 =

log
10
[H+]
[H+]= 10

3
How do you convert (back) from Vinegar’s concentration of hydrogen ions [H+] to
pH?
Take the log of both sides.
YOU TRY IT! Imagine that you have a body of water that has been
contaminated by acid r
ain. The pH is about equal to that of Vinegar. If
you were to dilute the water with an equal amount of water that had the
pH of Ammonia, would the result be a neutral pH level?
To answer the question above, compute the acidity, in terms of
concentration
of hydrogen ions and in terms of pH.
I cannot average the two pH’s, because pH represents an exponent.
Ammonia:
pH = 12
12‽

log
10
[H+]
†
孈[崽]10

12
Vinegar:
pH = 3
3‽

log
10
[H+]
†
[䠫崽10

3
The average concentration of H+ ions for a mixture
of equal parts
ammonia and vinegar is:
(
10

12
+ 10

3
)/2 = 0.0005
The pH associated with the mixture is: pH =

log(0.0005) = 3.3
*notice that the pH is much closer to vinegar than to ammonia.
Hint: remember that pH is on a logarithmic scale, so the val
ues of pH
are actually exponents.
7
The Decibel
Decibel
is a logarithmic unit used to describe the ratio of the signal level

power, sound pressure, voltage or intensity or several other
things.
www.armacell.com/.../$file/decibel

scale_227.jpg
,
http://www.engineeringtoolbox.
com/sound

power

intensity

pressure

d_57.html
Sound Pressure Level
The
Sound Pressure
(p) is the force
(N) of sound on a surface area (m
2
)
perpendicular to the direction of the sou
nd. The SI

units are N/m
2
or Pa.
L
p
= 10 log( p
2
/ p
ref
2
) = 10 log( p / p
ref
)
2
= 20 log ( p / p
ref
)
L
p
= sound pressure level (dB)
p = sound pressure (Pa)
p
ref
= 2 x 10

5

reference sound pressure (Pa)
QUESTION:
What is the sound pressure associated
with a 110 dB pneumatic drill?
110 = 20log(p/
2 x 10

5
)
p = 6.32 P
a
QUESTION:
Find the decibel level associated with a
sound pressure of 0.1 Pa. Mark your
answer on the decibel scale, to the left.
p = 0.1 Pa
p
ref
=
2 x 10

5
Pa
Lp = 20log(
0.1 Pa/
2 x 10

5
Pa)
Lp = 74 dB
8
Sound Power Level
Sound power is
the energy
rate

the energy of sound per unit of time (J/s, W in
SI

units) from a sound source.
Sound power can more practically be expressed as a relation to the threshold of
hearing

10

12
W

in a logarithmic scale named Sound Power Level

L
w
:
L
w
= 10 log (
N / No)
where
L
w
= Sound Power Level in
Decibel
(dB)
N = sound power (W)
The lowest sound level
that people of excellent hearing can discern has an
acoustic sound power about 10

12
W, 0 Db. This is the value used for
No.
TASK #1:
show that 10

12
W is, in fact, 0 dB using the equation for sound power
level.
L
w
= 10 log (
10

12
W
/
10

12
W
)
= 10 log (1)
= 0 (note: log(1) = 0)
TASK #2: FILL IN THE CHART BELOW
\
TASK #3
How
many order of magnitudes larger is the noise pain threshold (140 db) than the
noise from a nearby engine airplane?
TWO orders of magnitude larger
9
General rules of Exponents and Logarithms
Logs are the opposite of exponentiation
They are inverse functi
ons of each other
o
Just like “The square root function is the
inverse
of the square function.”
o
They undo each other
Linear vs. Exponential Growth
Some common situations where this happens:
Situation:
No
tes:
Population growth
Population increases by the same fraction (# people/persons)
in each time period.
Radioactive decay
Material decays at the same rate over and over
Compound interest
Money increases by a fixed percentage over and over
Industrial C
apital
Machines and factories generate new
machines and factories over and over
*Fill in the blank for above with your own “situation”!
Key Rules for Logarithms
Notes:
x
x
10
log
10
Definition of log
x
y
x
y
10
10
log
log
Logs turn exponentiation into multiplica
tion…
lets you solve for exponent!!!
1
10
log
10
Any number to the first power equals itself.
Use exponents to “undo” logs
=
Is the function linear, or exponential?
A linear function has a constant absolute rate of change.
An exponential function has a constant relative (or percent) rate of change.
Any situation in which the same thing ha
ppens over and over again,
and increases or decreases each time by a fixed percentage is
exponential growth or decay. ~Cooper, page 107
10
Algebraic Rules of Exponents and Logarithms (for reference)
Rules for log base 10. Note: base could be replaced with any other positive number
(such as
‘e’ or ‘2’)
except
0 or 1, and rules will still hold true. The letters ‘x’ and ‘y’ are arbitrary (they
could be replaced with any other letters), and they do NOT stand for independent/dependent
variables. ARGUMENTS IN EXPONENTS AND LOGARITHMS M
UST BE
DIMENSIONLESS.
EXAMPLES:
Where
e = 2.71828
Rules for Exponents
Rules for Logarithms
Notes:
....
10
*
10
*
10
10
x
(repeat x times)
x
x
10
log
10
Definition of log
y
x
y
x
10
10
10
y
x
xy
10
10
10
log
log
log
Logs turn multiplication into addition
y
x
y
x
10
10
10
y
x
y
x
10
10
10
log
log
log
Logs turn division into
subtraction.
y
x
y
x
*
10
10
x
y
x
y
10
10
log
log
Logs turn exponentiation into
multiplication… lets you solve for
exponent!!!
x
x
10
1
10
x
x
1
log
)
log(
10
Look a lot like the division to
subtraction rule…..
=
1
10
0
=
0
1
log
10
=
䅮y畭扥爠瑯⁴桥⁺e牯r灯睥爠r煵a汳l
潮o.
=
10
10
1
=
1
10
log
10
=
䅮y畭扥爠瑯⁴桥楲獴⁰潷o爠r煵a汳l
楴獥汦l
=
)
log(
)
(
log
10
y
y
x
means
x
y
10
y
x
b
log
means
y
b
x
)
ln(
)
(
log
y
y
x
e
means
x
e
y
11
Exponential Growth
N(t) = population (# of individuals... widgets... etc.) at time t
N
0
= population at time t = 0; initial condition
r = fractional growth rate (per unit time)
t = time
EXAMPLE:
California currently has a population of about 37 million. If the state continues to grow at the
current rate (about 1.2% per year), how many people will there be in 10 years?
p = 1.2%/yr
r = 0.12 1/yr
t = 10 yr
N
0
= 37 x 10
6
N
t
=10
= 3.7 x 10
7
e
0.012 (10)
N
t =10
= 42 million
In what year will there be twice as many people as there are now (74 million people)?
N
t
= N
0
e
rt
74x10
6
= 37x10
6
e
0.012t
2
= e
0.012t
ln 2
= ln(e
0.012t
)
t = (ln2)/0.012
t = 57.8 yrs
rt
e
N
t
N
0
)
(
12
Summary of Critical
Values for Exponential Growth
Value of r in Exponential Model:
Population is:
r > 0
Growing
r = 0
Constant; not growing or declining
r < 0
Declining
READING LOG SCALES
The parameters N
0
and r can be calculated using a semi

log plot
, as follows:.
N
0
can be read directly. The lines between “10” and “100” are in increments
of 10, while the lines between 100 and 1000 are in increments of 100... and so
forth
The rate of change (r) is calculated by choosing two points ON THE LINE
a
nd using the following equation:
rt
e
N
t
N
0
)
(
13
Note: “ln” is used instead of “log” because the exponential growth formula uses the base “e”.
Alternatively, you can calculate the slope using log (base 10) and convert your answer using
the rela
tionship
log
10
x = 0.434 ln(x).
TASK #1: What was the world oil production in the 1880?
About 40 Mbbl/yr
TASK #2: What is the rate of increase (r) in world oil production?
r =
ln(2000)
–
ln(40
) = 0.0674
58

0
TASK #3 Write down your
final estimated equation for N(t). Compare your
answer to the one on page 73 (estimated by Excel).
P(t) = 40 e
0.0674t
This is close to the equation estimated by Excel, which was P(t) =
40.81 e
0.0663t
14
CIA world factbook:
https://www.cia.gov/cia/publications/factbook/index.html
Country
Current Population
r
(% growth
rate per year)
r (fractional
growth rate
per year)
Pop. In 10 years (if r is
constant)
Pop. In 50 years
(if
r is constant)
India
1,095,351,995
N
o
= 1.1x10
9
1.38%
0.0138
1.25 x 10
9
people
1.1 x 10
9
e
(.0138)(10)
= 1.26x10
9
people
1.1
x
10
9
e
(.0138)(
5
0)
= 2.19x10
9
people
Gaza Strip
1,428,757
N
o
= 1.
4
x10
6
3.71%
.0371
1.4
x 10
6
e
(.0371
)(10)
= 2.02x10
6
people
1.4
x
10
6
e
(.0371)(5
0)
= 8.9x10
6
people
Germany
82,422,299
N
o
=
8.2
x10
7
0%
8.2
x10
7
people
(doesn’t change)
8.2
x10
7
people
(doesn’t change)
售匮
㈹㠬㐴㐬㈱2
N
o
=
3.0
x10
8
0.91%
.0091
3.0
x 10
8
e
(.0091
)(10)
= 3.2x10
8
people
3.0
x 10
8
e
(.0091)(5
0)
= 4.7
x10
8
people
Russia
142,893,540
N
o
= 1.
4
x10
8

0.37%

0.0037
1.4
x 10
8
e
(

.0037
)(10)
= 1.35x10
8
people
1.4
x 10
8
e
(

.0037)(5
0)
= 1.1x10
8
people
How much larger is the population of India than the population of the U.S., currently?
NO CALCULATORS (use scien
tific notation and rules of exponents).
1.1 x 10
9
= 1.1/3 x 10
9

8
≈ .33 x 10
1
= 3.3 times larger
3.0 x 10
8
How much larger will the population of India be compared to the population of the
U.S., in 50 years from now? NO CALCULATORS!!
2.19 x 10
9
= 2.2/4.4 x 10
9

8
= .5 x 10
1
= 5 times larger
4.7 x 10
8
15
Common A
pplications: Doubling Time and Half

Life
DOUBLING TIME
:
the doubling time of an exponentially increasing function is the time
that is take for the quantity to double.
Consider the continuous model of population growth:
rt
e
N
t
N
0
)
(
When will
the population double?
o
In other words, at what time, t, will the population be 2 times the original, or
initial, population size? ? Let’s call this time “doubling time”.
Let doubling time = T
d
N(T
d
) = 2N
0
Population at ‘doubling time’ is twice the i
nitial population, by definition
2 N
0
= N
0
e
r*Td
2 = e
r*Td
What if we used log instead of ln?
ln(2) = r*T
d
ln(e)
r
T
d
)
2
ln(
Rule of Thumb:
p
T
d
70
Where p = percent growth rate
p = 100 * r
Find the doubl
ing time for India’s population.
T
d
= 70/1.38 = 50.7 years
Does the doubling time depend on the initial (or current) population level?
No, the initial population does not matter, only the growth rate
does.
16
How many years will it take for
the population of the Gaza Strip to :
o
Double?
T
d
= 70/3.71 = 18.9 years
o
Triple?
N(T
t
) = 3No = N
o
e
rTt
ln3 = lne
rTt
ln3 = rT
t
T
t
= ln3/r = 110/r
T
t
= 110/3.71 = 29.6 years
o
Quadruple?
T
Q
= ln4/r = 138/r
T
Q
= 138/3.71 = 37.2 years
17
YOU
DERIVE
IT!!!
HALF

LIFE
: the half life of an exponentially decaying quantity is the time it takes for the
quantity to be reduced by a factor of one half.
Certain isotopes of elements with unstable nuclei spontaneously decay. The rate of decay is
propo
rtional to the amount of radioactive material. This phenomenon is very important for
applications such as nuclear fallout and carbon dating. The equation for radioactive decay is
given by:
kt
be
y
(compare this with the population growth
model)
y = amount of radioactive material at time t.
b = initial amount of radioactive material
k = decay constant (units are per unit time); this varies from one element to another.
The time necessary for half of the radioactive material to decay
is known as the half

life
of the material. Find the half

life of C
14
. The decay constant (k) is known to be 1.24 x
10

4
per year [HINT: the amount of radioactive material will be ½ of the original
amount at the time corresponding to the half

life].
*
Check your answer by doing a quick google search of ‘carbon 14 half life’.
C
14
half life:
50 = 100e
–
(1.24x10

4)(t)
.5 = e
–
kt
ln.5 = lne
–
kt
ln.5 =
–
kt
ln.5 =
–
(1.24x10

4
)t
5590 = t
½ life of C
14
= 5590 years
*According to the web it’s 5
730 years
–
the difference is actually
due to the decay constant (k) being a little off. Try solving for k,
given a ½ life of C
14
of 5730 years.
18
The principle of the carbon

dating method is as follows: The proportion of the radioactive isotope
C
14
to the regular isotope C
12
of carbon occurring in the earth’s atmosphere remains at a constant
level, the amount of C
14
that decays being exactly balanced by new C
14
, which is formed by
cosmic rays hitting the atmosphere. Living plants and animals absorb
carbon from the atmosphere,
and so contain a proportion of C
14
that is more or less the same as that in the atmosphere. When
the animal or vegetation dies, however, the absorption of new C
14
ceases, and the proportion of C
14
in the dead organic matter st
eadily decreases as a result of radioactive decay. By measuring the
proportion of C
14
to C
12
in organic matter obtained from an archaeological site, it is possible to
calculate how long it is since the death of the animal or vegetable occurred. [Arya, Mat
hematics
for the Biological Sciences, page 155]
A human bone is measured to have 58% of the radioactive carbon that occurs naturally in
the atmosphere. How old is the bone? (given: k
(C14)
= 1.24 x 10

4
)
58 = 100e
–
(1.24x10

4)(t)
.58 = e
–
(1.24x10

4)(t)
l
n.58 = lne
–
(1.24x10

4)(t)
–
.5447=
–
(1.24x10

4
)t
t = 4393 years
The level of radioactivity on the site of a nuclear explosion is decaying exponentially. The
level measured in 1990 was found to be 0.7 times the level measured in 1980. What is
the
half

life? [Hint: first solve for k, then solve for the half

life]
70 = 100e
–
k(10)
ln.7 =
–
10k*lne
–
.3567 =
–
10k
.0357 = k
50 =
100e
–
.0357t
ln.5 =
–
.0357t*lne
–
.6931 =
–
.0357t
½ life: 19.4 yrs
19
Exponential growth in action: EARNING INTEREST and
DISCOUNTING
EARNING INTEREST
(see Cooper, pg 107)
n
i
P
A
)
1
(
where:
A = total amount after ‘n’ compounding periods ($)
P = the principal, or initial amount of money ($)
i =
Annual Interest Rate (fractional)
(unitless)
Numb
er of Compoundings per year
n = total number of compounding periods for the life of the loan (unitless)
Example 1:
If you invest $100 in the bank which pays you interest at a rate 10%, compounded
monthly (12 times per year), how much money wil
l you have after 10 years?
A = 100 (1 + .10/12)
120
A = $270.70
Example 2:
How much is $1 worth in 1 year, if it earns an annual interest rate of 100%, and is
only compounded once (at the end of the year)?
A = 1 (1 + 1/1)
1
A = $2
What if it is compound
ed monthly, over a one year period, instead?
A = 1 (1 + 1/12)
12
A = $2.61
What if it is compounded each minute, over a one year period?
A = 1 (1 + 1/525,600)
525,600
A = $2.72
20
Exponential decay in action: DISCOUNTING
Discounting
is the
opposite
of
earni
ng interest
Present Value answers the question: “How much money would you set aside now to
avoid a cost in the future?”
o
Example: How much will we spend now to avoid damages from global
warming that will not occur until 50 years from now?
Present Value a
nswers the question: “How much money would you set aside now to get
benefits in the future?”
o
Example: How much are we willing to invest in renewable energy that will not
provide us with benefits until 10 years from now?
Present Value decays exponentiall
y; the rate of decay is our ‘discount rate’
How much is a future value of 1 million dollars worth today?
You are thinking of investing in a new solar power company. You are told that you
will receive an income of $ 1 million dollars in 10 years (becaus
e it takes time to set up
the infrastructure, establish the business, etc). How much will you invest now to get
this future benefit?
Using a 3% discount rate, you would invest up to $744,000. If you
use a higher discount rate, say 10%, you would only be w
illing to
invest $386,000.
21
What if you wont receive the benefit for 50 years, instead of 10 years?
Using a 3% discount rate, you would invest up to $228,000.
Using a 3% discount rate, you would invest up to $8500.
Future Value in year ‘n’:
PV
d
FV
n
*
)
1
(
(does this look familiar?)
Present Value:
n
d
FV
PV
)
1
(
Where
PV = present value ($)
FV= future value ($)
n = number of years from present (assuming annual compounding)
d = discount rate (fractional)
Note: in these formulas
, the present year is year zero (n = 0)!
22
YOU TRY IT!!!
Calculate the present value of 1 million dollars received in 10 years, using a
discount rate of 10 %.
x =
1,000,000
= $385,543
(1 + 0.1)
10
We will play around with these graphs in Excel!
POP QUIZ: Circle one:
Humans prefer benefits:
now
later
Humans prefer costs:
now
later
To Ponder: Does discounting conflict with ‘sustainability’?
23
EXPONENTS AND LOGS: KEY CONCEPTS
Exponents and logs are used to deal with numbe
rs that vary by orders of magnitude
(powers of 10)
Logs are used to “undo” exponentiation
Natural log means a logarithm with base
e
o
e
= 2.71828… it is a magic number (like pi)
On a graph, a log scale is used to view data that varies by orders of magni
tude.
Any situation in which the same things happen over and over again, and increases or
decreases each time by a fixed percentage is exponential growth or decay..
o
An exponential function has a constant relative (or percent) rate of change.
o
Contrast wi
th: A linear function has a constant absolute rate of change.
Doubling time (know rule of thumb)
Half life for exponential decay
Applications of exponential functions
o
Population growth
o
Earning Interest
o
Discounting
Understand how this accounting pract
ice devalues future environmental
benefits
24
HOMEWORK PROBLEMS
1.
Solve for x.
a.
log
10
(3.162) = x
x = 0.5
b.
e
x
= 100 Show your work.
x = 4.6
2. Earning Interest
(refer to your workbook for formulas):
A bank pays 7% interest compounded ann
ually. You put $5000 in the account in January
2007. In what year will there be 1 million dollars in the account? Show your work.
PV
d
FV
n
*
)
1
(
10
6
= (1.07)
n
*5000
n = 78.3 years
3. Discounting:
If you are still confused about discounting,
please re

read “The Binding
Value of Time,” by Garrett Hardin (reader pgs 63

72).
Calculation: (refer to your workbook for formulas):
a.
You will receive $200,000 in 30 years. Calculate the present value of this payment,
using a 7% discount rate. Show your
work.
n
d
FV
PV
)
1
(
PV =
200,000
= $26, 273
(1.07)
30
b.
Explain what your answer means in a way that would be clear to someone who has
never hear of discounting.
I would only be willing to give up $26, 273 now to receive $200,000
in 30 ye
ars. Or, I would be willing to pay $26, 273 now to avoid
paying $200,000 in 30 years. Both of these statements are based on
a 7% discount rate.
25
READ the following excerpt from:
A Look at Nature's Numbers, by John Gibbons,
National Academy of Engineerin
g
ANSWER the questions that follow.
To start off, I'd like to draw a couple of verbal images. "Eat, drink and be merry, for tomorrow
you may die." That's from the Rubaiyat of Omar Khayyam. Now, here's one I heard during the
Reagan administration when I as
ked a person about some future issues, and he said, "Why
should I worry about the future? What's the future done for me?" Sort of a modern Rubaiyat, I
guess.
…
We live in an age of discount rates. I think we all pretty much know how to calculate the net
pr
esent value of future things. We depreciate buildings and other things that decay and in a
sense go to zero value at some point
[examples?
]
, but we
also seem to be willing to use some
of those same principles for evaluations of things like biological speci
es for which it's very
difficult to conceptualize how you can depreciate them to zero over a period of time.
We live
in a time in which a typical corporate manager has to worry about, not next year's profit, but
next quarter's! So we operate in a time of e
xtraordinarily high discount rates in terms of the
present value of future conditions.
Here's another vignette. When archeologists excavated in Russell Cave, Alabama, and found
some of the earliest artifacts of human presence in North America, they purpos
efully left
untouched a major portion of that cave in which surely lie some very important artifacts. They
left it alone for future generations because they knew that technology would likely advance
over the years and that a much better excavation could be
done 50 or 100 years down the road.
A different kind of sense of discount rates and preparation for the future
.
[What discount rate
would you use? Positive, negative, or zero?]
.
We have, within the last 10 or 20 years, begun to think very seriously about
such things as
natural capital. There's a recent Academy publication called Nature's Numbers (authored by
some distinguished economists) in which we're now beginning to wrestle with the fact that
there are goods and services in our economy, the value of w
hich have never been incorporated
into our national economic accounts (our way of accounting goods and services delivered to
people). These are so

called "natural capital" accounts, such as the natural environment that
cleans water, that provid
es fertiliza
tion of crops, all the other so

called services provided by
natural ecosystems. We're now in the process of trying to figure out how we can link them into
our economic reporting rather than leaving them outside the systems of national accounts
.[do
you thin
k this is a good idea? Are there alternatives to putting a $ value on the
environment?]
Even rough measures tell us that a very substantial portion of our wealth
comes from outside our economic system as traditionally calculated.
So we're in the middle of
a
very interesting transition of realizing the sources of wealth and our
responsibilities to the future for not destroying that wealth without at least putting something in
place of it
.
It has been brought to the forefront by a man/biosphere crisis that
has emerged in
the 20th century as a result of rapid population growth and rapid industrialization, and it is on a
collision course in the 21st century. There are clear mandates, it seems to me, for us to
26
understand this business and think again about what
the stewardship responsibilities are for
humans.
These issues are presently being ignored by the public in general, by business, by politicians
—
where political lifetimes are very short. You know, when a congressman gets elected, he must
immediately start
campaigning for the next election. President Clinton told me once when we
were working on a climate protection protocol, and I argued for a 20

to 30

year time horizon,
"You're absolutely right about the need for a long time horizon, but no number greater
than 10
years has any consequence in politics. The discount rate wipes you out."
Follow

up Questions:
1. Why do politicians tend to have a high discount rate? Does this conflict with the
purpose of government (What is the purpose of government..?
)
Some reasons: Short political terms to get re

elected, need economy to see
SHORT TERM gains, as long

term stability will be on some other
politician’s watch… thus they are more concerned with the PRESENT than
the future.
I think that the role of gover
nment is protect rights and preserve justice, and
to serve the needs of the people. If future peoples’ rights and needs count,
discounting does seem to be at odds with the role of government.
2. How does discounting play into everyday decisions, such as
:
Pretty much any situation where you enjoy now, and pay later
credit cards: you want the benefit NOW, even though you will have to
pay a higher price later
purchase of durable vs. disposable goods
o
Disposable: cheap now, but will have to pay for it
again in the
future
o
Durable: more expensive, but will last longer
.
3.
Personal example of discounting
Describe a situation where you
“discounted the future” in your everyday life (no need to feel
guilty, all humans discount, that is why the formula, or
model of human behavior, exists).
See answer above
27

choosing NOT to buy rechargeable batteries or energy efficient
appliances, buying cheap, low quality goods instead of more
expensive, high quality goods (such as clothes/furniture/cars/etc.)
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