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PART 1

a) Write a history on logarithm.

History of Logarithms

Logarithms were invented independently by John Napier, a Scotsman, and by Joost Burgi, aSwiss. Napier's logarithms were published in 1614; Burgi's logarithms were published in 1620.The objective of both men was to simplify mathematical calculations. This approach originallyarose out of a desire to simplify multiplication and division to the level of addition andsubtraction. Of course, in this era of the cheap hand calculator, this is not necessary anymore butit still serves as a useful way to introduce logarithms. Napier's approach was algebraic andBurgi's approach was geometric. The invention of the common system of logarithms is due to thecombined effort of Napier and Henry Biggs in 1624. Natural logarithms first arose as more or less accidental variations of Napier's original logarithms. Their real significance was notrecognized until later. The earliest natural logarithms occur in 1618.

It cant be said too often: a logarithm is nothing more than an exponent. The basic concept of logarithms can be expressed as a shortcut.. Multiplication is a shortcut for Addition: 3 x 5 means 5 + 5 + 5Exponents are a shortcut for Multiplication: 4^3 means 4 x 4 x 4Logarithms are a shortcut for Exponents: 10^2 = 100.

The present definition of the logarithm is the exponent or power to which a stated number, calledthe base, is raised to yield a specific number.The logarithm of 100 to the base 10 is 2. This iswritten: log10 (100) = 2. Before pocket calculators only th ree decades ago, but in studentyears thats the age of dinosaurs the answer was simple. You needed logs to compute most

powers and roots with fair accuracy; even multiplying and dividing most numbers were easier with logs. Every decent algebra books had pages and pages of log tables at the back. Theinvention of logs in the early 1600s fueled the scientific revolution. Back then scientists,astronomers especially, used to spend huge amounts of time crunching numbers on paper. Bycutting the time they spent doing arithmetic, logarithms effectively gave them a longer

productive life. The slide rule, once almost a cartoon trademark of a scientist, was nothing morethan a device built for doing various computations quickly, using logarithms.

http://en.wikipedia.org/wiki/File:John_Napier.jpg
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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. Fractals

The Sierpinski triangle (at the right) is constructed by repeatedly replacing equilateraltriangles by three smaller ones.

Logarithms occur in definitions of the dimension of fractals . Fractals are geometric objectsthat ar eself-similar : small parts reproduce, at least roughly, the entire global structure.The Sierpinski triangle ( pictured) can be covered by three copies of itself, each having sideshalf the original length. This makes the Hausdorff dimension of this structure log(3)/log(2) 1.58. Another logarithm-based notion of dimension is obtained by counting the number of

boxes needed to cover the fractal in question.

2. Number theory Natural logarithms are closely linked to counting prime numbers (2, 3, 5, 7, 11, ...), animportant topic in number theory . For any integer x, the quantity of prime numbers less thanor equal to x is denoted ( x). The prime number theorem asserts that ( x) is approximatelygiven by

in the sense that the ratio of ( x) and that fraction approaches 1 when x tends to infinity. As aconsequence, the probability that a randomly chosen number between 1 and x is prime isinversely proportional to the numbers of decimal digits of x. A far better estimate of ( x) isgiven by the offset logarithmic integral function Li( x), defined by

The Riemann hypothesis , one of the oldest open mathematical conjectures , can be stated interms of comparing ( x) and Li( x). The Erds Kac theorem describing the number of distinct prime factors also involves the natural logarithm.

The logarithm of n factorial , n! = 1 2 ... n, is given by

This can be used to obtain Stirling's formula , an approximation of n! for large n.

http://en.wikipedia.org/wiki/Equilateral_trianglehttp://en.wikipedia.org/wiki/Equilateral_trianglehttp://en.wikipedia.org/wiki/Equilateral_trianglehttp://en.wikipedia.org/wiki/Fractal_dimensionhttp://en.wikipedia.org/wiki/Fractal_dimensionhttp://en.wikipedia.org/wiki/Fractal_dimensionhttp://en.wikipedia.org/wiki/Fractalhttp://en.wikipedia.org/wiki/Fractalhttp://en.wikipedia.org/wiki/Fractalhttp://en.wikipedia.org/wiki/Self-similarityhttp://en.wikipedia.org/wiki/Self-similarityhttp://en.wikipedia.org/wiki/Self-similarityhttp://en.wikipedia.org/wiki/Sierpinski_trianglehttp://en.wikipedia.org/wiki/Sierpinski_trianglehttp://en.wikipedia.org/wiki/Sierpinski_trianglehttp://en.wikipedia.org/wiki/Hausdorff_dimensionhttp://en.wikipedia.org/wiki/Hausdorff_dimensionhttp://en.wikipedia.org/wiki/Hausdorff_dimensionhttp://en.wikipedia.org/wiki/Box-counting_dimensionhttp://en.wikipedia.org/wiki/Box-counting_dimensionhttp://en.wikipedia.org/wiki/Box-counting_dimensionhttp://en.wikipedia.org/wiki/Box-counting_dimensionhttp://en.wikipedia.org/wiki/Prime-counting_functionhttp://en.wikipedia.org/wiki/Prime-counting_functionhttp://en.wikipedia.org/wiki/Prime-counting_functionhttp://en.wikipedia.org/wiki/Number_theoryhttp://en.wikipedia.org/wiki/Number_theoryhttp://en.wikipedia.org/wiki/Number_theoryhttp://en.wikipedia.org/wiki/Integerhttp://en.wikipedia.org/wiki/Integerhttp://en.wikipedia.org/wiki/Integerhttp://en.wikipedia.org/wiki/Prime_numberhttp://en.wikipedia.org/wiki/Prime_numberhttp://en.wikipedia.org/wiki/Prime_numberhttp://en.wikipedia.org/wiki/Prime-counting_functionhttp://en.wikipedia.org/wiki/Prime-counting_functionhttp://en.wikipedia.org/wiki/Prime-counting_functionhttp://en.wikipedia.org/wiki/Prime-counting_functionhttp://en.wikipedia.org/wiki/Prime-counting_functionhttp://en.wikipedia.org/wiki/Prime_number_theoremhttp://en.wikipedia.org/wiki/Prime_number_theoremhttp://en.wikipedia.org/wiki/Prime_number_theoremhttp://en.wikipedia.org/wiki/Proportionality_(mathematics)http://en.wikipedia.org/wiki/Proportionality_(mathematics)http://en.wikipedia.org/wiki/Proportionality_(mathematics)http://en.wikipedia.org/wiki/Logarithmic_integral_functionhttp://en.wikipedia.org/wiki/Logarithmic_integral_functionhttp://en.wikipedia.org/wiki/Logarithmic_integral_functionhttp://en.wikipedia.org/wiki/Riemann_hypothesishttp://en.wikipedia.org/wiki/Riemann_hypothesishttp://en.wikipedia.org/wiki/Riemann_hypothesishttp://en.wikipedia.org/wiki/Conjecturehttp://en.wikipedia.org/wiki/Conjecturehttp://en.wikipedia.org/wiki/Conjecturehttp://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Kac_theoremhttp://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Kac_theoremhttp://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Kac_theoremhttp://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Kac_theoremhttp://en.wikipedia.org/wiki/Prime_factorhttp://en.wikipedia.org/wiki/Prime_factorhttp://en.wikipedia.org/wiki/Prime_factorhttp://en.wikipedia.org/wiki/Factorialhttp://en.wikipedia.org/wiki/Factorialhttp://en.wikipedia.org/wiki/Factorialhttp://en.wikipedia.org/wiki/Stirling%27s_formulahttp://en.wikipedia.org/wiki/Stirling%27s_formulahttp://en.wikipedia.org/wiki/Stirling%27s_formulahttp://en.wikipedia.org/wiki/File:Sierpinski_dimension.svghttp://en.wikipedia.org/wiki/File:Sierpinski_dimension.svghttp://en.wikipedia.org/wiki/File:Sierpinski_dimension.svghttp://en.wikipedia.org/wiki/File:Sierpinski_dimension.svghttp://en.wikipedia.org/wiki/Stirling%27s_formulahttp://en.wikipedia.org/wiki/Factorialhttp://en.wikipedia.org/wiki/Prime_factorhttp://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Kac_theoremhttp://en.wikipedia.org/wiki/Conjecturehttp://en.wikipedia.org/wiki/Riemann_hypothesishttp://en.wikipedia.org/wiki/Logarithmic_integral_functionhttp://en.wikipedia.org/wiki/Proportionality_(mathematics)http://en.wikipedia.org/wiki/Prime_number_theoremhttp://en.wikipedia.org/wiki/Prime-counting_functionhttp://en.wikipedia.org/wiki/Prime_numberhttp://en.wikipedia.org/wiki/Integerhttp://en.wikipedia.org/wiki/Number_theoryhttp://en.wikipedia.org/wiki/Prime-counting_functionhttp://en.wikipedia.org/wiki/Box-counting_dimensionhttp://en.wikipedia.org/wiki/Box-counting_dimensionhttp://en.wikipedia.org/wiki/Hausdorff_dimensionhttp://en.wikipedia.org/wiki/Sierpinski_trianglehttp://en.wikipedia.org/wiki/Self-similarityhttp://en.wikipedia.org/wiki/Fractalhttp://en.wikipedia.org/wiki/Fractal_dimensionhttp://en.wikipedia.org/wiki/Equilateral_trianglehttp://en.wikipedia.org/wiki/Equilateral_triangle
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PART 2

The volume, V, in cm 3, 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 diameter of the 6spheres 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, cm 3.

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find the volume of sphere using water displacement method.

A method of finding the volume of a sphere with minimal calculations is to use the Water

Displacement Method :

1. Fill a beaker or graduated cylinder with enough water to completely immerse the spherein.

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 V 1= 0.5D2 =2.5 V 2= 8.0D3 =3.6 V 3= 25.0D4 =5.0 V 4= 65.0D5 =6.8 V 5= 165.0

D6 =7.8 V 6= 248.0

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Diameter,D ( Volume, V (

D1 = 1.0 V 1= 0.5D2 =2.5 V 2= 8.0D3 =3.6 V 3= 25.0D4 =5.0 V 4= 65.0D

5=6.8 V

5= 165.0

D6 =7.8 V 6= 248.0

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 V 1= 0.5D6 =7.8 V 6= 248.0

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Therefore, and

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PART 3

3(A)

D v

1 0.5

2.8 8

4 25

5.2 65

6.6 165

7.8 248

y = 0.504x 3.0228

0

50

100

150

200

250

300

0 1 2 3 4 5 6 7 8 9

V o

l u m e

, V

Diameter, D

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3(B)

y = 3.0228x - 0.2976

-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

l o g V

Diameter, D

log D log V0 -0.30103

0.447158 0.903090.60206 1.39794

0.716003 1.812913

0.819544 2.2174840.892095 2.394452

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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 = 3.0228x - 0.2976

-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

l o g V

Diameter, D

log D log V

0 -0.301030.447158 0.90309

0.60206 1.397940.716003 1.8129130.819544 2.2174840.892095 2.394452

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2. Volume of the sphere when diameter is 5cmSince 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

FURTHER EXPLORATION

y = 3.0228x - 0.2976

-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

l o g V

log D

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a) -------(1) ------------(2)

(1)=(2)

D

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

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

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REFLECTION

Symbols used in this project using Microsoft word equation insert tool really help me somuch 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 = 0.4277x 3.0596

0

50

100

150

200

250

300

0 2 4 6 8 10

V o

l u m e

, V

Diameter, D

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