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PART 1

a) Write a history on logarithm.

History of Logarithms

Predecessors

TheBabylonianssometime in 20001600 BC may have invented thequarter square

multiplicationalgorithm to multiply two numbers using only addition, subtraction and a

table of squares. However it could not be used for division without an additional table of

reciprocals. Large tables of quarter squares were used to simplify the accurate

multiplication of large numbers from 1817 onwards until this was superseded by the use

of computers.

Michael StifelpublishedArithmetica integra inNurembergin 1544, which contains a

table of integers and powers of 2 that has been considered an early version of a

logarithmic table.

In the 16th and early 17th centuries an algorithm calledprosthaphaeresiswas used to

approximate multiplication and division. This used the trigonometric identity

or similar to convert the multiplications to additions and table lookups. However

logarithms are more straightforward and require less work. It can be shown using

complex numbers that this is basically the same technique.

From Napier to Euler

John Napier (15501617), the inventor of logarithms

The method of logarithms was publicly propounded byJohn Napierin 1614, in a booktitled Mirifici Logarithmorum Canonis Descriptio(Description of the Wonderful Rule of

Logarithms).Joost Brgiindependently invented logarithms but published six years

after Napier.

Johannes Kepler, who used logarithm tables extensively to compile his Ephemeris and

therefore dedicated it to Napier, remarked:

http://en.wikipedia.org/wiki/Babylonian_mathematicshttp://en.wikipedia.org/wiki/Babylonian_mathematicshttp://en.wikipedia.org/wiki/Babylonian_mathematicshttp://en.wikipedia.org/wiki/Multiplication_algorithm#Quarter_square_multiplicationhttp://en.wikipedia.org/wiki/Multiplication_algorithm#Quarter_square_multiplicationhttp://en.wikipedia.org/wiki/Multiplication_algorithm#Quarter_square_multiplicationhttp://en.wikipedia.org/wiki/Multiplication_algorithm#Quarter_square_multiplicationhttp://en.wikipedia.org/wiki/Michael_Stifelhttp://en.wikipedia.org/wiki/Michael_Stifelhttp://en.wikipedia.org/wiki/Nuremberghttp://en.wikipedia.org/wiki/Nuremberghttp://en.wikipedia.org/wiki/Nuremberghttp://en.wikipedia.org/wiki/Prosthaphaeresishttp://en.wikipedia.org/wiki/Prosthaphaeresishttp://en.wikipedia.org/wiki/Prosthaphaeresishttp://en.wikipedia.org/wiki/John_Napierhttp://en.wikipedia.org/wiki/John_Napierhttp://en.wikipedia.org/wiki/John_Napierhttp://en.wikipedia.org/wiki/Joost_B%C3%BCrgihttp://en.wikipedia.org/wiki/Joost_B%C3%BCrgihttp://en.wikipedia.org/wiki/Joost_B%C3%BCrgihttp://en.wikipedia.org/wiki/Johannes_Keplerhttp://en.wikipedia.org/wiki/Johannes_Keplerhttp://en.wikipedia.org/wiki/File:John_Napier.jpghttp://en.wikipedia.org/wiki/File:John_Napier.jpghttp://en.wikipedia.org/wiki/Johannes_Keplerhttp://en.wikipedia.org/wiki/Joost_B%C3%BCrgihttp://en.wikipedia.org/wiki/John_Napierhttp://en.wikipedia.org/wiki/Prosthaphaeresishttp://en.wikipedia.org/wiki/Nuremberghttp://en.wikipedia.org/wiki/Michael_Stifelhttp://en.wikipedia.org/wiki/Multiplication_algorithm#Quarter_square_multiplicationhttp://en.wikipedia.org/wiki/Multiplication_algorithm#Quarter_square_multiplicationhttp://en.wikipedia.org/wiki/Babylonian_mathematics
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...the accent in calculation led Justus Byrgius [Joost Brgi] on the way to these

very logarithms many years before Napier's system appeared; but ...instead of

rearing up his child for the public benefit he deserted it in the birth.Johannes Kepler, Rudolphine Tables (1627)

By repeated subtractions Napier calculated (1 107)L forL ranging from 1 to 100. The

result forL=100 is approximately0.99999 = 1 105. Napier then calculated the

products of these numbers with 107(1 105)L forL from 1 to 50, and did similarly

with0.9998 (1 105)20 and 0.9 0.99520. These computations, which occupied 20

years, allowed him to give, for any numberNfrom 5 to 10 million, the numberL that

solves the equation

Napier first called L an "artificial number", but later introduced the word "logarithm"to

mean a number that indicates a ratio: (logos) meaning proportion,

and (arithmos) meaning number. In modern notation, the relation to natural

logarithms is

where the very close approximation corresponds to the observation that

The invention was quickly and widely met with acclaim. The works

ofBonaventura Cavalieri(Italy),Edmund Wingate(France), Xue Fengzuo

(China), andJohannes Kepler's Chilias logarithmorum (Germany) helped spread

the concept further.

In 1647Grgoire de Saint-Vincentrelated logarithms to the quadrature of the

hyperbola, by pointing out that the area f(t) under the hyperbola fromx=

1 tox= tsatisfies

The natural logarithm was first described byNicholas Mercatorin his

work Logarithmotechnia published in 1668, although the mathematics teacher John

Speidell had already in 1619 compiled a table on the natural logarithm. Around

1730,Leonhard Eulerdefined the exponential function and the natural logarithm by

Euler also showed that the two functions are inverse to one another.

http://en.wikipedia.org/wiki/Logoshttp://en.wikipedia.org/wiki/Logoshttp://en.wikipedia.org/wiki/Logoshttp://en.wikipedia.org/wiki/Bonaventura_Cavalierihttp://en.wikipedia.org/wiki/Bonaventura_Cavalierihttp://en.wikipedia.org/wiki/Bonaventura_Cavalierihttp://en.wikipedia.org/wiki/Edmund_Wingatehttp://en.wikipedia.org/wiki/Edmund_Wingatehttp://en.wikipedia.org/wiki/Edmund_Wingatehttp://en.wikipedia.org/wiki/Johannes_Keplerhttp://en.wikipedia.org/wiki/Johannes_Keplerhttp://en.wikipedia.org/wiki/Johannes_Keplerhttp://en.wikipedia.org/wiki/Gr%C3%A9goire_de_Saint-Vincenthttp://en.wikipedia.org/wiki/Gr%C3%A9goire_de_Saint-Vincenthttp://en.wikipedia.org/wiki/Gr%C3%A9goire_de_Saint-Vincenthttp://en.wikipedia.org/wiki/Nicholas_Mercatorhttp://en.wikipedia.org/wiki/Nicholas_Mercatorhttp://en.wikipedia.org/wiki/Nicholas_Mercatorhttp://en.wikipedia.org/wiki/Leonhard_Eulerhttp://en.wikipedia.org/wiki/Leonhard_Eulerhttp://en.wikipedia.org/wiki/Leonhard_Eulerhttp://en.wikipedia.org/wiki/Nicholas_Mercatorhttp://en.wikipedia.org/wiki/Gr%C3%A9goire_de_Saint-Vincenthttp://en.wikipedia.org/wiki/Johannes_Keplerhttp://en.wikipedia.org/wiki/Edmund_Wingatehttp://en.wikipedia.org/wiki/Bonaventura_Cavalierihttp://en.wikipedia.org/wiki/Logos
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b) Find and explain the applications of logarithm in two different fields of study.Explanation of each application should include the following

I. The field of study chosen.II. Examples of problem solving related to the field of study

Application of Logarithms

1. Psychology

Logarithms occur in several laws describinghuman perception:Hick's lawproposes a

logarithmic relation between the time individuals take for choosing an alternative and

the number of choices they have.Fitts's lawpredicts that the time required to rapidly

move to a target area is a logarithmic function of the distance to and the size of the

target. Inpsychophysics, theWeberFechner lawproposes a logarithmic relationship

betweenstimulusandsensationsuch as the actual vs. the perceived weight of an item

a person is carrying. (This "law", however, is less precise than more recent models,

such as theStevens' power law.)

Psychological studies found that individuals with little mathematics education tend to

estimate quantities logarithmically, that is, they position a number on an unmarked line

according to its logarithm, so that 10 is positioned as close to 100 as 100 is to 1000.

Increasing education shifts this to a linear estimate (positioning 1000 10x as far away) in

some circumstances, while logarithms are used when the numbers to be plotted are

difficult to plot linearly.

2. Probability theory and statistics

Threeprobability density functions(PDF) of random variables with log-normal

distributions. The location parameter, which is zero for all three of the PDFs shown, is

the mean of the logarithm of the random variable, not the mean of the variable itself.

http://en.wikipedia.org/wiki/Human_perceptionhttp://en.wikipedia.org/wiki/Human_perceptionhttp://en.wikipedia.org/wiki/Human_perceptionhttp://en.wikipedia.org/wiki/Hick%27s_lawhttp://en.wikipedia.org/wiki/Hick%27s_lawhttp://en.wikipedia.org/wiki/Hick%27s_lawhttp://en.wikipedia.org/wiki/Fitts%27s_lawhttp://en.wikipedia.org/wiki/Fitts%27s_lawhttp://en.wikipedia.org/wiki/Fitts%27s_lawhttp://en.wikipedia.org/wiki/Psychophysicshttp://en.wikipedia.org/wiki/Psychophysicshttp://en.wikipedia.org/wiki/Weber%E2%80%93Fechner_lawhttp://en.wikipedia.org/wiki/Weber%E2%80%93Fechner_lawhttp://en.wikipedia.org/wiki/Weber%E2%80%93Fechner_lawhttp://en.wikipedia.org/wiki/Weber%E2%80%93Fechner_lawhttp://en.wikipedia.org/wiki/Weber%E2%80%93Fechner_lawhttp://en.wikipedia.org/wiki/Stimulus_(psychology)http://en.wikipedia.org/wiki/Stimulus_(psychology)http://en.wikipedia.org/wiki/Stimulus_(psychology)http://en.wikipedia.org/wiki/Sensation_(psychology)http://en.wikipedia.org/wiki/Sensation_(psychology)http://en.wikipedia.org/wiki/Sensation_(psychology)http://en.wikipedia.org/wiki/Stevens%27_power_lawhttp://en.wikipedia.org/wiki/Stevens%27_power_lawhttp://en.wikipedia.org/wiki/Stevens%27_power_lawhttp://en.wikipedia.org/wiki/Probability_density_functionhttp://en.wikipedia.org/wiki/Probability_density_functionhttp://en.wikipedia.org/wiki/Probability_density_functionhttp://en.wikipedia.org/wiki/File:Benfords_law_illustrated_by_world's_countries_population.pnghttp://en.wikipedia.org/wiki/File:Benfords_law_illustrated_by_world's_countries_population.pnghttp://en.wikipedia.org/wiki/File:Some_log-normal_distributions.svghttp://en.wikipedia.org/wiki/File:Some_log-normal_distributions.svghttp://en.wikipedia.org/wiki/File:Benfords_law_illustrated_by_world's_countries_population.pnghttp://en.wikipedia.org/wiki/File:Benfords_law_illustrated_by_world's_countries_population.pnghttp://en.wikipedia.org/wiki/File:Some_log-normal_distributions.svghttp://en.wikipedia.org/wiki/File:Some_log-normal_distributions.svghttp://en.wikipedia.org/wiki/File:Benfords_law_illustrated_by_world's_countries_population.pnghttp://en.wikipedia.org/wiki/File:Benfords_law_illustrated_by_world's_countries_population.pnghttp://en.wikipedia.org/wiki/File:Some_log-normal_distributions.svghttp://en.wikipedia.org/wiki/File:Some_log-normal_distributions.svghttp://en.wikipedia.org/wiki/File:Benfords_law_illustrated_by_world's_countries_population.pnghttp://en.wikipedia.org/wiki/File:Benfords_law_illustrated_by_world's_countries_population.pnghttp://en.wikipedia.org/wiki/File:Some_log-normal_distributions.svghttp://en.wikipedia.org/wiki/File:Some_log-normal_distributions.svghttp://en.wikipedia.org/wiki/Probability_density_functionhttp://en.wikipedia.org/wiki/Stevens%27_power_lawhttp://en.wikipedia.org/wiki/Sensation_(psychology)http://en.wikipedia.org/wiki/Stimulus_(psychology)http://en.wikipedia.org/wiki/Weber%E2%80%93Fechner_lawhttp://en.wikipedia.org/wiki/Psychophysicshttp://en.wikipedia.org/wiki/Fitts%27s_lawhttp://en.wikipedia.org/wiki/Hick%27s_lawhttp://en.wikipedia.org/wiki/Human_perception
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Distribution of first digits (in %, red bars) in thepopulation of the 237 countriesof the

world. Black dots indicate the distribution predicted by Benford's law.

Logarithms arise inprobability theory: thelaw of large numbersdictates that, for afair

coin, as the number of coin-tosses increases to infinity, the observed proportion of

described by thelaw of the iterated logarithm.

Logarithms also occur inlog-normal distributions. When the logarithm of arandom

variablehas anormal distribution, the variable is said to have a log-normal

distribution. Log-normal distributions are encountered in many fields, wherever a

variable is formed as the product of many independent positive random variables, for

example in the study of turbulence.

Logarithms are used formaximum-likelihood estimationof parametricstatistical models.

For such a model, thelikelihood functiondepends on at least oneparameterthat must

be estimated. A maximum of the likelihood function occurs at the same parameter-value

as a maximum of the logarithm of the likelihood (the " log likelihood"), because thelogarithm is an increasing function. The log-likelihood is easier to maximize, especially

for the multiplied likelihoods forindependentrandom variables.

Benford's lawdescribes the occurrence of digits in manydata sets, such as heights of

buildings. According to Benford's law, the probability that the first decimal-digit of an

item in the data sample is d(from 1 to 9) equals log10(d+ 1) log10(d), regardless of the

unit of measurement. Thus, about 30% of the data can be expected to have 1 as first

digit, 18% start with 2, etc. Auditors examine deviations from Benford's law to detect

fraudulent accounting.

http://en.wikipedia.org/wiki/List_of_countries_by_populationhttp://en.wikipedia.org/wiki/List_of_countries_by_populationhttp://en.wikipedia.org/wiki/List_of_countries_by_populationhttp://en.wikipedia.org/wiki/Probability_theoryhttp://en.wikipedia.org/wiki/Probability_theoryhttp://en.wikipedia.org/wiki/Probability_theoryhttp://en.wikipedia.org/wiki/Law_of_large_numbershttp://en.wikipedia.org/wiki/Law_of_large_numbershttp://en.wikipedia.org/wiki/Law_of_large_numbershttp://en.wikipedia.org/wiki/Fair_coinhttp://en.wikipedia.org/wiki/Fair_coinhttp://en.wikipedia.org/wiki/Fair_coinhttp://en.wikipedia.org/wiki/Fair_coinhttp://en.wikipedia.org/wiki/Binomial_distribution#Symmetric_binomial_distribution_.28p_.3D_0.5.29http://en.wikipedia.org/wiki/Binomial_distribution#Symmetric_binomial_distribution_.28p_.3D_0.5.29http://en.wikipedia.org/wiki/Binomial_distribution#Symmetric_binomial_distribution_.28p_.3D_0.5.29http://en.wikipedia.org/wiki/Law_of_the_iterated_logarithmhttp://en.wikipedia.org/wiki/Law_of_the_iterated_logarithmhttp://en.wikipedia.org/wiki/Law_of_the_iterated_logarithmhttp://en.wikipedia.org/wiki/Log-normal_distributionhttp://en.wikipedia.org/wiki/Log-normal_distributionhttp://en.wikipedia.org/wiki/Log-normal_distributionhttp://en.wikipedia.org/wiki/Random_variablehttp://en.wikipedia.org/wiki/Random_variablehttp://en.wikipedia.org/wiki/Random_variablehttp://en.wikipedia.org/wiki/Random_variablehttp://en.wikipedia.org/wiki/Normal_distributionhttp://en.wikipedia.org/wiki/Normal_distributionhttp://en.wikipedia.org/wiki/Normal_distributionhttp://en.wikipedia.org/wiki/Maximum-likelihood_estimationhttp://en.wikipedia.org/wiki/Maximum-likelihood_estimationhttp://en.wikipedia.org/wiki/Maximum-likelihood_estimationhttp://en.wikipedia.org/wiki/Statistical_modelhttp://en.wikipedia.org/wiki/Statistical_modelhttp://en.wikipedia.org/wiki/Statistical_modelhttp://en.wikipedia.org/wiki/Likelihood_functionhttp://en.wikipedia.org/wiki/Likelihood_functionhttp://en.wikipedia.org/wiki/Parametric_modelhttp://en.wikipedia.org/wiki/Parametric_modelhttp://en.wikipedia.org/wiki/Parametric_modelhttp://en.wikipedia.org/wiki/Independence_(probability)http://en.wikipedia.org/wiki/Independence_(probability)http://en.wikipedia.org/wiki/Independence_(probability)http://en.wikipedia.org/wiki/Benford%27s_lawhttp://en.wikipedia.org/wiki/Benford%27s_lawhttp://en.wikipedia.org/wiki/Data_sethttp://en.wikipedia.org/wiki/Data_sethttp://en.wikipedia.org/wiki/Data_sethttp://en.wikipedia.org/wiki/Data_sethttp://en.wikipedia.org/wiki/Benford%27s_lawhttp://en.wikipedia.org/wiki/Independence_(probability)http://en.wikipedia.org/wiki/Parametric_modelhttp://en.wikipedia.org/wiki/Likelihood_functionhttp://en.wikipedia.org/wiki/Statistical_modelhttp://en.wikipedia.org/wiki/Maximum-likelihood_estimationhttp://en.wikipedia.org/wiki/Normal_distributionhttp://en.wikipedia.org/wiki/Random_variablehttp://en.wikipedia.org/wiki/Random_variablehttp://en.wikipedia.org/wiki/Log-normal_distributionhttp://en.wikipedia.org/wiki/Law_of_the_iterated_logarithmhttp://en.wikipedia.org/wiki/Binomial_distribution#Symmetric_binomial_distribution_.28p_.3D_0.5.29http://en.wikipedia.org/wiki/Fair_coinhttp://en.wikipedia.org/wiki/Fair_coinhttp://en.wikipedia.org/wiki/Law_of_large_numbershttp://en.wikipedia.org/wiki/Probability_theoryhttp://en.wikipedia.org/wiki/List_of_countries_by_population
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PART 2

The volume, V, in cm3, of a solid sphere and its diameter, D, in cm, are related by the

equation , where m and n are constants.

Find the value of m and n by conducting the activities below.

I. Choose 6 different spheres with diameters between 1cm to 8cm. The diameterof the 6 spheres using a pair of vernier calipers.

II. Find the volume of each sphere using water displacement method.III. Tabulate the values of diameter, D, in cm and its corresponding volume, V, cm3.

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find the volume of sphere using water displacement menthod.

A method of finding the volume of a sphere with minimal calculations is to use the WaterDisplacement Method:

1. Fill a beaker or graduated cylinder with enough water to completely immerse thesphere in.

2. Record the baseline initial measurement3. Drop the sphere in4. Record final measurement5. Subtract the initial volume from the final volume ~ this is the volume of the

sphere!

Value of diameter,D and Volume

Diameter,D ( Volume, V (

D1 = 1.0 V1= 0.5

D2 =2.2 V2= 5.5

D3 =3.5 V3= 23.0

D4 =4.8 V4= 58.0

D5 =6.5 V5= 142.0

D6 =8.0 V6= 268.0

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Diameter,D ( Volume, V (

D1 = 1.0 V1= 0.5

D2 =2.2 V2= 5.5

D3 =3.5 V3= 23.0

D4 =4.8 V4= 58.0

D5=6.5 V

5= 142.0

D6 =8.0 V6= 268.0

We can solve by simultaneous method

Substitute the values in the equation

We obtain,

----------(1)

----------(2)

-----------(3)

Substitute (3) into (2)

-----------(4)

Substitute (4) into (3)

D1 = 1.0 V1= 0.5

D3 =3.5 V3= 23.0

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Therefore, and

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PART 3

3(A)

D v

1 0.5

2.2 5.5

3.5 23

4.8 58

6.5 142

8 268

y = 0.5071x3.0193

0

50

100

150

200

250

300

0 1 2 3 4 5 6 7 8 9

Volume,V

Diameter, D

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3(B)

y = 3.0193x - 0.2949

-0.5

0

0.5

1

1.5

2

2.5

3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

logV

Diameter, D

log D log V0 -0.30103

0.342423 0.740363

0.544068 1.361728

0.681241 1.763428

0.812913 2.152288

0.90309 2.428135

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3c) From the graph, find

1. The value of m and of n, thus express V in terms of D.

(nearest whole number)

y = 3.0193x - 0.2949

-0.5

0

0.5

1

1.5

2

2.5

3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

logV

Diameter, D

log D log V

0 -0.30103

0.342423 0.7403630.544068 1.361728

0.681241 1.763428

0.812913 2.152288

0.90309 2.428135

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2. Volume of the sphere when diameter is 5cm

Since graph is logV against logD, we need to transfer, D=5cm int0

logD=log5=0.6989

We get

3. The radius of the sphere when the volume is

Change to logv=log180=2.25,

From the graph, we get

y = 3.0193x - 0.2949

-0.5

0

0.5

1

1.5

2

2.5

3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

logV

Diameter, D

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FURTHER EXPLORATION

a) -------(1)

------------(2)

(1)=(2)

D

-------------------cancel on both sides

b) Another method to find value of is using Monte Carlo simulation or

Archimedes method of Exhaustion

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REFLECTION

Symbols used in this project using Microsoft word equation insert tool really help me so

much here are some of the symbol I use.

I really learn how to use Microsoft excel to do graph, insert equation and a lot more.

y = 0.5071x3.0193

0

50

100

150

200

250

300

0 2 4 6 8 10

Volume,V

Diameter, D

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