# Exponential and Logarithmic Functions

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4 Δεκ 2013 (πριν από 4 χρόνια και 5 μήνες)

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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

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:

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)

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+]

-
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
-
12

W, 0 Db. This is the value used for
No.

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

\

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.

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

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

=
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

=
)
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

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?

TASK #2: What is the rate of increase (r) in world oil production?

r =
ln(2000)

ln(40
) = 0.0674

58
-

0

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)

㈹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

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

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].

*

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
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

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,
-
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

A Look at Nature's Numbers, by John Gibbons,

g

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
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

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).