AddMaths Project Work

7/28/2019 AddMaths Project Work

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LogarithmName : AHMAD ASYRAAF BIN SHAHRIZAMAN

Class : 5 SN 2

NRIC : 961106-10-6763

Teachers Name : PN. WAN MASTURA BT WAN ABBAS

School: SMK Tinggi Klang

Additional Mathematics Project 1/2013


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Num Content Page

1 Acknowledgement 1

2 Objective 2

3 Introduction 3

4 Part 1 4-7

5 Part 2 8

6 Part 3 9-12

7 Further Exploration 13-15

8 Reflection 16-17

9 Reference 18

Additional Mathematics Project 1/2013


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First and foremost I would like to thank to Allah that finally, I have succeeded in

finishing this project work. I would like to thank my beloved Additional Mathematics teacher,

Puan Wan Mastura for all the guidance she had provided me during the process of finishing

this work.

Next, I would like to give a thousand appreciations to both En Shahrizaman and Pn

Roszela as my parent who had gave me full support in this project, financially , facilities and

mentally. They gave me moral support when I needed it.

I also would like to give my thanks to my fellow friends such as Ahmad Anwar and Hairi

Adzimuddin who had helped me in finding the information that Im clueless about, and the

time we spent together in study groups on finishing this project during school holidays. Last

but not least I would like to express this appreciation towards all who gave me the

possibility to complete this coursework.

1 Additional Mathematics Project 1/2013


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The aims of carrying out project work are :

To apply and adapt a variety of problem- solving strategies to solve problems

To improve thinking skills

To promote effective mathematical communication

To develop mathematical knowledge through problem solving in a way that

increases students interest and confidence

To use language of mathematics to express mathematical ideas precisely

To provide learning environment that stimulates and enchances effective learning

To develop positive attitude towards mathematics



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One of the mathematical concepts which we must be master up is logarithms.Before

the days of scientific calculators,logarithms were used to multiply or divide extreme

numbers using mathematical tables.For these calculations, ten was the most common base

to use.Secondary students usually use base 10 to solve calculation by using

calculator.Logarithm to the base of ten is also called the common logarithm that can be

writen wihout state its base (log ).Other bases such as two,five and eight can be also

used.The ancient Babylonians had used bases up to 60.

Logarithms have many applications in various fields of studies.In the early 17th

century it

rapidly adopted by navigators, scientists, chemists, engineers and astronomers to perfom

computations more easily.



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a) History of Logarithms

Logarithms were invented independently by John Napier, a Scotsman, and by Joost Burgi, a Swiss.

Napier's logarithms were published in 1614; Burgi's logarithms were published in 1620. Theobjective of both men was to simplify mathematical calculations. This approach originally arose out

of a desire to simplify multiplication and division to the level of addition and subtraction. Of course,

in this era of the cheap hand calculator, this is not necessary anymore but it still serves as a useful

way to introduce logarithms. Napier's approach was algebraic and Burgi's approach was geometric.

The invention of the common system of logarithms is due to the combined 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 not recognized 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 oflogarithms can be expressed as a shortcut.. Multiplication is a shortcut for Addition: 3 x 5 means

5 + 5 + 5 Exponents are a shortcut for Multiplication: 4^3 means 4 x 4 x 4 Logarithms are a shortcut

for Exponents: 102

= 100.

The present definition of the logarithm is the exponent or power to which a stated number,

called the base, is raised to yield a specific number. The logarithm of 100 to the base 10 is 2. This is

written: log10 (100) = 2.

Before pocket calculators only three decades ago, but in student years 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.

The invention 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. By

cutting 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 more than a device

built for doing various computations quickly, using logarithms. See Eli Maors e: The Story of a

Number for more on this. Today, logs are no longer used in routine number crunching. But there are

still good reasons for studying them.



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b) Application of logarithm in fields of study

In Chemistry,logarithm is used to measure the pH or acidity of a chemical solution. The pH

is the negative logarithm of the concentration of free hydrogen ions. pH is a logarithmic

measure of hydrogen ion concentration, originally defined by Danish biochemist

Sren Peter Lauritz Srensen in 1909 [1].

pH= -log [ H+

]

p H (Concentration of H+

ions)

where log is a base-10 logarithm and [H+] is the concentration of hydrogen ions inmoles per liter of solution. According to the Compact Oxford English Dictionary, the"p" stands for the German word for "power", potenz, so pH is an abbreviation for"power of hydrogen" [2].

The pH scale was defined because the enormous range of hydrogen ionconcentrations found in aqueous solutions make using H+ molarity awkward. Forexample, in a typical acid-base titration, [H+] may vary from about 0.01 M to0.0000000000001 M. It is easier to write "the pH varies from 2 to 13".

Example of problem solving :

1.What is the pH of an aqueous solution when the concentration of hydrogen ion is5.0 x 10-4 M?

pH = -log [H+]

= -log (5.0 x 10-4)

= - (-3.30)

= 3.30 #

2.What is the concentration of the hydrogen ion concentration in an aqueous solutionwith pH = 13.22?

pH = -log [H+] = 13.22log [H+] = -13.22

[H+] = inv log (-13.22)[H+] = 6.0 x 10-14 M # (2 sig. fig.)

http://antoine.frostburg.edu/chem/senese/101/acidbase/faq/what-is-pH.shtml#1http://antoine.frostburg.edu/chem/senese/101/acidbase/faq/what-is-pH.shtml#1http://antoine.frostburg.edu/chem/senese/101/acidbase/faq/what-is-pH.shtml#1http://antoine.frostburg.edu/chem/senese/101/acidbase/faq/what-is-pH.shtml#2http://antoine.frostburg.edu/chem/senese/101/acidbase/faq/what-is-pH.shtml#2http://antoine.frostburg.edu/chem/senese/101/acidbase/faq/what-is-pH.shtml#2http://antoine.frostburg.edu/chem/senese/101/acidbase/faq/what-is-pH.shtml#2http://antoine.frostburg.edu/chem/senese/101/acidbase/faq/what-is-pH.shtml#1


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InElectronics, gain is a measure of the ability of acircuit(often anamplifier) to increasethepoweroramplitudeof asignalfrom the input to the output, by adding energy to the signal

converted from somepower supply. It is usually defined as the meanratioof thesignal

outputof a system to thesignal inputof the same system. It may also be defined on a

logarithmic scale, in terms of the decimallogarithmof the same ratio ("dBgain"). A gain

greater than one (zero dB), that is, amplification, is the defining property of anactivecomponentor circuit, while apassive circuitwill have a gain of less than one.

Thus, the termgain on its own is ambiguous. For example, "a gain of five" may imply that

either thevoltage,currentor thepoweris increased by a factor of five, although most often

this will mean a voltage gain of five for audio and general purposeamplifiers, especially

operational amplifiers, but a power gain forradio frequencyamplifiers. Furthermore, the term

gain is also applied in systems such assensorswhere the input and output have different units;

in such cases the gain units must be specified, as in "5 microvolts per photon" for the

responsivityof a photosensor. The "gain" of abipolar transistornormally refers to forward

current transfer ratio, eitherhFE ("Beta", the static ratio ofIc divided byIb at some operating

point), or sometimes hfe (the small-signal current gain, the slope of the graph ofIc againstIbata point).

The termgain has a slightly different meaning inantennadesign;antenna gainis the ratio of

power received by a directional antenna to power received by anisotropic antenna.

Logarithmic units and decibels formula

Power gain

Voltage gain

http://en.wikipedia.org/wiki/Electronicshttp://en.wikipedia.org/wiki/Electronicshttp://en.wikipedia.org/wiki/Electronicshttp://en.wikipedia.org/wiki/Electrical_networkhttp://en.wikipedia.org/wiki/Electrical_networkhttp://en.wikipedia.org/wiki/Electrical_networkhttp://en.wikipedia.org/wiki/Amplifierhttp://en.wikipedia.org/wiki/Amplifierhttp://en.wikipedia.org/wiki/Amplifierhttp://en.wikipedia.org/wiki/Power_(physics)http://en.wikipedia.org/wiki/Power_(physics)http://en.wikipedia.org/wiki/Power_(physics)http://en.wikipedia.org/wiki/Amplitudehttp://en.wikipedia.org/wiki/Amplitudehttp://en.wikipedia.org/wiki/Amplitudehttp://en.wikipedia.org/wiki/Signal_(electrical_engineering)http://en.wikipedia.org/wiki/Signal_(electrical_engineering)http://en.wikipedia.org/wiki/Signal_(electrical_engineering)http://en.wikipedia.org/wiki/Power_supplyhttp://en.wikipedia.org/wiki/Power_supplyhttp://en.wikipedia.org/wiki/Power_supplyhttp://en.wikipedia.org/wiki/Ratiohttp://en.wikipedia.org/wiki/Ratiohttp://en.wikipedia.org/wiki/Ratiohttp://en.wikipedia.org/wiki/Signalling_(telecommunication)http://en.wikipedia.org/wiki/Signalling_(telecommunication)http://en.wikipedia.org/wiki/Signalling_(telecommunication)http://en.wikipedia.org/wiki/Signalling_(telecommunication)http://en.wikipedia.org/wiki/Signalling_(telecommunication)http://en.wikipedia.org/wiki/Signalling_(telecommunication)http://en.wikipedia.org/wiki/Signalling_(telecommunication)http://en.wikipedia.org/wiki/Logarithmhttp://en.wikipedia.org/wiki/Logarithmhttp://en.wikipedia.org/wiki/Logarithmhttp://en.wikipedia.org/wiki/Decibelhttp://en.wikipedia.org/wiki/Decibelhttp://en.wikipedia.org/wiki/Decibelhttp://en.wikipedia.org/wiki/Active_componenthttp://en.wikipedia.org/wiki/Active_componenthttp://en.wikipedia.org/wiki/Active_componenthttp://en.wikipedia.org/wiki/Active_componenthttp://en.wikipedia.org/wiki/Passive_circuithttp://en.wikipedia.org/wiki/Passive_circuithttp://en.wikipedia.org/wiki/Passive_circuithttp://en.wikipedia.org/wiki/Voltagehttp://en.wikipedia.org/wiki/Voltagehttp://en.wikipedia.org/wiki/Voltagehttp://en.wikipedia.org/wiki/Electric_currenthttp://en.wikipedia.org/wiki/Electric_currenthttp://en.wikipedia.org/wiki/Electric_currenthttp://en.wikipedia.org/wiki/Electric_powerhttp://en.wikipedia.org/wiki/Electric_powerhttp://en.wikipedia.org/wiki/Electric_powerhttp://en.wikipedia.org/wiki/Amplifierhttp://en.wikipedia.org/wiki/Amplifierhttp://en.wikipedia.org/wiki/Amplifierhttp://en.wikipedia.org/wiki/Operational_amplifierhttp://en.wikipedia.org/wiki/Operational_amplifierhttp://en.wikipedia.org/wiki/Radio_frequencyhttp://en.wikipedia.org/wiki/Radio_frequencyhttp://en.wikipedia.org/wiki/Radio_frequencyhttp://en.wikipedia.org/wiki/Sensorhttp://en.wikipedia.org/wiki/Sensorhttp://en.wikipedia.org/wiki/Sensorhttp://en.wikipedia.org/wiki/Responsivityhttp://en.wikipedia.org/wiki/Responsivityhttp://en.wikipedia.org/wiki/Bipolar_transistorhttp://en.wikipedia.org/wiki/Bipolar_transistorhttp://en.wikipedia.org/wiki/Bipolar_transistorhttp://en.wikipedia.org/wiki/Antenna_(radio)http://en.wikipedia.org/wiki/Antenna_(radio)http://en.wikipedia.org/wiki/Antenna_(radio)http://en.wikipedia.org/wiki/Antenna_gainhttp://en.wikipedia.org/wiki/Antenna_gainhttp://en.wikipedia.org/wiki/Antenna_gainhttp://en.wikipedia.org/wiki/Isotropic_antennahttp://en.wikipedia.org/wiki/Isotropic_antennahttp://en.wikipedia.org/wiki/Isotropic_antennahttp://en.wikipedia.org/wiki/Isotropic_antennahttp://en.wikipedia.org/wiki/Antenna_gainhttp://en.wikipedia.org/wiki/Antenna_(radio)http://en.wikipedia.org/wiki/Bipolar_transistorhttp://en.wikipedia.org/wiki/Responsivityhttp://en.wikipedia.org/wiki/Sensorhttp://en.wikipedia.org/wiki/Radio_frequencyhttp://en.wikipedia.org/wiki/Operational_amplifierhttp://en.wikipedia.org/wiki/Amplifierhttp://en.wikipedia.org/wiki/Electric_powerhttp://en.wikipedia.org/wiki/Electric_currenthttp://en.wikipedia.org/wiki/Voltagehttp://en.wikipedia.org/wiki/Passive_circuithttp://en.wikipedia.org/wiki/Active_componenthttp://en.wikipedia.org/wiki/Active_componenthttp://en.wikipedia.org/wiki/Decibelhttp://en.wikipedia.org/wiki/Logarithmhttp://en.wikipedia.org/wiki/Signalling_(telecommunication)http://en.wikipedia.org/wiki/Signalling_(telecommunication)http://en.wikipedia.org/wiki/Signalling_(telecommunication)http://en.wikipedia.org/wiki/Ratiohttp://en.wikipedia.org/wiki/Power_supplyhttp://en.wikipedia.org/wiki/Signal_(electrical_engineering)http://en.wikipedia.org/wiki/Amplitudehttp://en.wikipedia.org/wiki/Power_(physics)http://en.wikipedia.org/wiki/Amplifierhttp://en.wikipedia.org/wiki/Electrical_networkhttp://en.wikipedia.org/wiki/Electronics


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Current gain

Example of problem solving :

Q.An amplifier has an input impedance of 50 ohms and drives a load of 50 ohms. When its

input ( ) is 1 volt, its output ( ) is 10 volts. What is its voltage and power gain?

A.Voltage gain is simply:

The units V/Vare optional, but make it clear that this figure is a voltage gain and not a power

gain. Using the expression for power,P= V2/R, the power gain is:

Again, the units W/W are optional. Power gain is more usually expressed in decibels, thus:

A gain of factor 1 (equivalent to 0 dB) where both input and output are at the same voltage

level and impedance is also known asunitygain.

http://en.wikipedia.org/wiki/1_(number)http://en.wikipedia.org/wiki/1_(number)http://en.wikipedia.org/wiki/1_(number)http://en.wikipedia.org/wiki/1_(number)


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a b

/cm /cm31.56 2.0

1.85 3.3

2.52 8.4

2.82 11.7

4.63 52.0

5.31 78.4

L


= lg =lg + lglg =lg+ lg

When

=156,

=20 lg20 =lg

+ lg156 ------ 1When =252, =84 lg84 =lg + lg252 ------ 2

= ooo ----------------- 3

lg84 =lg + *ooo lg252 ------------Subtitute 3 into 2lg84 =lg +2078 lg22078lg

1078 lg

=2078 lg2lg84

lg = oo =05283 # (4 sig. fig. )


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[A] Plot graph of against (refer to graph paper)/cm 0.0 1.6 1.9 2.5 2.8 4.6 5.3/cm3 0.0 2.0 3.3 8.4 11.7 52.0 78.4



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[B] a) Reduce the equation = to a linear formlg = lg + lg lg = lg + lg lg = lg + lg

b) Plot graph lg against lg (refer to graph paper)lg 0.19 0.27 0.40 0.45 0.67 0.73lg 0.30 0.52 0.92 1.06 1.72 1.89

c) Gradient, = =2.9

=024i) Gradient,m = =29 #

=lg=024=05754

= 05754 ii) When

= 5 , = 057545

= 6123 iii) = 05248 = 2

When =180, 180= 05754

=2

2 = (

)

=3626



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a) Equation obtained form Part 3 [B] c (i) is = 05754Volume of sphere, =

= 05754 = 2 =

=(05754

= (0.5754)

2

If = 1 , 1 =0575421=3221



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b) There is many method to find the value of.The method that can be use is :Method 1

Measure the circumference of circle,

.It can be measured by using thread. Then,

measure its diameter,.Diameter is equivalent to double times of radius,r ( = 2 )

So that, find the value of by substitute the value of and into formula: = 2 =

= We can conclude that the value of is the ratio of Circumference, to its Diameter,.



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Method 2

Find the volume of object, such as cyclinder by using water-displacement method.Theobject is immersed in water and volume of water displaced is measured.According to

physics principle,the volume of water displaced is equal to the volume of object.Theprinciple is called Archimedes Principle.

From the diagram,the volume of glass rod immersed in water, is equal to final waterlevel, initial water level, = .The value of and its radius,r and also itsheight, is substitute into the formula:

=

=



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While I conducting this project,I find out a lot of information.As an example,I find out how

to find the pH of the acid without using pH meter that I havent learn in school.This

increases my knowledge.I also learn how to use apparatus in lab and the way of interpreting

data.Now I understand that logarithm is useful to be applied in modern daily life.We should

thankful to mathematican for discover logarithm.It is not easy as a piece of cake.I know that

mathematics is interesting to be discover.

I also learn some moral values that I practice.This project taught me to be responsible

and discipline on the works given by teacher to be completed in three weeks.I felt

confidence and do not give up to find solution in solving many kind of problems during

conducting the project.This project also learn me the value of friendship.I did this project

with friends and cooperate in collecting and interpreting data.I came to school with my

friends to complete this task during school holidays.This tighten our friendship.



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Websites link :

http://en.wikipedia.org/wiki/Logarithm http://en.wikipedia.org/wiki/Gain

http://www.youtube.com/watch?v=lca_puB1R8k

http://www.chem.tamu.edu/class/fyp/mathrev/mr-log.html

http://en.wikipedia.org/wiki/Pi

http://en.wikipedia.org/wiki/Logarithmhttp://en.wikipedia.org/wiki/Logarithmhttp://en.wikipedia.org/wiki/Gainhttp://en.wikipedia.org/wiki/Gainhttp://www.youtube.com/watch?v=lca_puB1R8http://www.chem.tamu.edu/class/fyp/mathrev/mr-log.htmlhttp://www.chem.tamu.edu/class/fyp/mathrev/mr-log.htmlhttp://en.wikipedia.org/wiki/Pihttp://en.wikipedia.org/wiki/Pihttp://en.wikipedia.org/wiki/Pihttp://www.chem.tamu.edu/class/fyp/mathrev/mr-log.htmlhttp://www.youtube.com/watch?v=lca_puB1R8http://en.wikipedia.org/wiki/Gainhttp://en.wikipedia.org/wiki/Logarithm

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