Exponential and Logarithmic Functions

ALGEBRIC MATHEMATICAL OPERATIONS

DIRECT OPERATIONS INVERSE OPERATIONS

a) addition subtractiona + b = c b = c – a

b) multiplication divisionab = c b = c/a

c) exponent or power logarithm

ba = c a = logb c

Logarithmic Functions

WHAT ARE LOGARITHMS AND WHY ARE THEY USEFUL?

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 numerical excesses in comprehensible terms.” It appears that human senses perceive the world though a logarithmic lens.

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A logarithmic rule of thumb:

For a number larger than one, written in ordinary decimal notation, a rough approximation of its logarithm can be obtained by counting its digits to the left of the decimal point. For a number written in standard scientific notation the exponent of the ten is roughly its logarithm.

Common log scales in Environmental Science:

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wwww.armacell.com/.../$file/decibel-scale_227.jpg

Sound SourceTotal Loudness

Power (P)Scientific Notation Log

dB = 10log(P)

Airplane Engine Nearby 1000000000000  Inside Airplane Cabin 10000000000  Pneumatic Drill 1000000000  Noisy Office 10000000  Ordinary Conversation 10000  Quiet Conversation 1000  Soft Rustle of Leaves 10  

Interesting Exercise

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Let’s see how how many decibels change occurs when we double a given loudness power. Suppose we have a guitar amp which is set to produce sound with P total loudness power. Then our formula gives us 10log(p) dB. Now suppose we crank the amp to double the loudness to 2P. What is the change in dB?

What is the loudness power that corresponds to 200 dB?

Exponential and Logarithmic FunctionsComposite graphic representation

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General rules

Logs are the opposite of exponentiation They are inverse functions of each other

o Just like “The square root function is the inverse of the square function.”o They undo each other

Video on Orders of Magnitude: http://micro.magnet.fsu.edu/primer/java/scienceopticsu/powersof10/

Some common situations where this happens:

Situation: Notes:Population growth Population (#births-#deaths) grows by the same amount in

each time period.Radioactive decay Material decays at the same rate over and overCompound interest Money increases by a fixed percentage over and over

*Fill in the blank for above with your own “situation”!

Rules for Exponents Rules for Logarithms Notes:....10*10*1010 =x

(repeat x times)xx =10log10 Definition of log

yxyx +=101010 yxxy 101010 logloglog += Logs turn multiplication into addition

yxy

x−=10

1010 yx

yx

101010 logloglog −= Logs turn division into subtraction.

( ) yxyx *1010 = xyx y 1010 loglog = Logs turn exponentiation into multiplication… lets you solve for exponent!!!

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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. ~Note Cooper, page 107

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.

xx

10110 =− ⎟⎠

⎞⎜⎝⎛=−x

x 1log)log( 10Look a lot like the division to subtraction rule…..

1100 = 01log10 = Any number to the zero power equals one.

10101 = 110log10 = Any number to the first power equals itself.

Rules for log base 10. Note: base could be replaced with any other number (such as ‘e’ or ‘2’), and rules will still hold true.. The letters ‘x’ and ‘y’ are arbitrary (they could be replaced with any other letters), they do NOT stand for independent/dependent variables. ARGUMENTS IN EXPONENTS AND LOGARITHMS MUST BE DIMENSIONLESS.

b is positive and not =1. b is called the ‘base’ Three important values for ‘b’ are 2 (computers), e (math & science), 10 (human

history, slide rule, etc)

EXAMPLES:

Where e = 2.71828… it’s a MAGIC #!!

To switch between bases, use:

bx

xa

ab log

loglog =

Examples:

log2x = 1.44 ln x log10x = 0.434 ln x

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)log()(log10 yyx == means xy 10=

yx blog= means

€

y = bx

)ln()(log yyx e == means xey =

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Exponential Population Growth

N(t) = population (# of individuals) at time t

N0 = population at time t = 0; initial condition

r = fractional growth rate (per unit time)

t = time

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 1.38% 0.0138 1.25 x 109 people

Gaza Strip 1,428,757 3.71%

Germany 82,422,299 0% 00000

U.S. 298,444,215 0.91%

Russia 142,893,540 -0.37% -0.0037

How much larger is the population of India than the population of the U.S., currently? NO CALCULATORS (use scientific notation and rules of exponents).

How much larger will the population of India be compared to the population of the U.S., in 50 years from now? NO CALCULATORS!!

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rteNtN 0)( =

Summary of Critical Values for Exponential Growth:

Value of r in Exponential Model: Population is:r > 0 Growingr = 0 Constant; not growing or decliningr < 0 Declining

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Use rules of logs to solve for an exponent:

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: rteNtN 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 = Td

N(Td) = 2N0 Population at ‘doubling time’ is twice the initial population, by definition

2 N0 = N0 e r*Td

2 = e r*Td What if we used log instead of ln?

ln(2) = r*Td ln(e)

rTd

)2ln(= Rule of Thumb: p

Td70

≈ Where p = percent growth rate

p = 100 * r

Find the doubling time for India’s population.

Does the doubling time depend on the initial (or current) population level?

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How many years will it take for the population of the Gaza Strip to :

o Double?

o Triple?

o Quadruple?

TRY WITH EXCEL! How long will it take for the population of the Gaza Strip (population = 1,428,757; r = 3.71%) to exceed the population of Israel (population= 6,352,117; r = 1.18%) assuming that population growth continues at current rates? This is most easily done in EXCEL (make columns for Time, Pop(Gaza), Pop(Israel); then click and drag the time and population formula boxes down until the population of Gaza is greater than the population of Israel. To work properly, you population formula must be linked to the time cell with a ‘relative’ reference.

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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 proportional 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:

ktbey −= (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 C14. 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’.

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The principle of the carbon-dating method is as follows: The proportion of the radioactive isotope C14 to the regular isotope C12 of carbon occurring in the earth’s atmosphere remains at a constant level, the amount of C14 that decays being exactly balanced by new C14, which is formed by cosmic rays hitting the atmosphere. Living plants and animals absorb carbon from the atmosphere, and so contain a proportion of C14

that is more or less the same as that in the atmosphere. When the animal or vegetation dies, however, the absorption of new C14 ceases, and the proportion of C14 in the dead organic matter steadily decreases as a result of radioactive decay. By measuring the proportion of C14 to C12 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, Mathematics 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)

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]

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Exponential growth in action: EARNING INTEREST (see Cooper, pg 107)

niPA )1( +=

where:

A = total amount after ‘n’ compounding periods ($)

P = the principal, or initial amount of money ($)

i = Annual Interest Rate (fractional) (unitless) Number 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 will you have after 10 years?

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)?

What if it is compounded monthly, over a one year period, instead?

What if it is compounded each minute, over a one year period?

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Exponential decay in action: DISCOUNTING

Discounting is the opposite of earning 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 answers 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 exponentially; 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 (because it takes time to set up the infrastructure, establish the business, etc). How much will you invest now to get this future benefit?

What if you wont receive the benefit for 50 years, instead of 10 years?

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Future Value in year ‘n’: PVdFV n *)1( += (does this look familiar?)

Present Value: ndFVPV

)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)!

YOU TRY IT!!! Calculate the present value of 1 million dollars received in 10 years, using a discount rate of 10 %.

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We will play around with these graphs in Excel!

POP QUIZ: Circle one:

Humans prefer benefits: now laterHumans prefer costs: now later

To Ponder: Does discounting conflict with ‘sustainability’?

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KEY CONCEPTS of Exponents and Logs

Exponents and logs are used to deal with numbers 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 magnitude.

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 with: 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 growtho Earning Interesto Discounting

Understand how this accounting practice devalues future environmental benefits

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