Math core project - term3

Introducing

Inventor Of Logarithms

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1Introduction 2Part one

3Part Two

Introduction

Dividing our work

1

The logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number. For example, the logarithm of 1000 to base 10 is 3, because 1000 is 10 to the power 3: 1000 = 103 = 10 × 10 × 10. More generally, if x = by, then y is the logarithm of x to base b, and is written logb(x), so log10(1000) = 3.

Getting Started With The Project

FYI:

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: Logarithms were introduced by John Napier in the early 17th century as a means to simplify calculations. They were rapidly adopted by scientists, engineers, and others to perform computations more easily, using slide rules and logarithm tables. These devices rely on the fact—important in its own right—that the logarithm of a product is the sum of the logarithms of the factors:

The present-day notion of logarithms comes from Leonhard Euler, who connected them to the exponential function in the 18th century.

The logarithm to base b = 10 is called the Common logarithms and has many applications in science and engineering. The natural logarithms has the constant e (≈ 2.718) as its base; its use is widespread in Pure mathematics, especially Calculus. The binary logarithm uses base b = 2 and is prominent in computer science.

2 Part One

Radiocarbon Dating

Questions 1:

By chemical analysis it has been determined that the amount of C-14 remaining in the samples of the Lascaux charcoal was 15% of the amount such trees would contain when living. The half-life of C-14 is approximately 5600 years. Write an equation to model this situation. (HINT: Write the equation of the decay then find the value of the constant k. Then you can estimate the time using the results from the first step.)

Solving Question1Equation:   y=a.e-Kt  Calculations  to  find  k:0.5a =ae-K(5600) both (a) will cancel each other -k.5600/-5600=In0.5/-5600 use the ( In) you will find it in the calculator   -k=In0.5/-5600= 0.000123776≈0.00012 Calculations  to  find  the  time  (t):      

0.15a= a.e-0.00012t t =In0.15-0.00012=15809.33≈15809year

Question 2:

The average cost of a home in Tucson is roughlyaround $225,000. Suppose you were planning to put down $25,000 now and finance the rest on a 30‐year mortgage at 7% compounded monthly. How much would your monthly payements be?Step 1 : The monthly payments. Step 2 : The sum of the numbers .

= 150.307568 ( as shown ) . So, F = 200,000 / sum : 150.307568 = 1330.60499

Step  3:    a.  Use  the  excel  sheet  to  graph  K  versus  Pk  .  What  happens  to   your  outstanding  balance  after  each  monthly  payment  is   made?                                 

b.  Is  this  a  graph  of  an  exponential  growth  or  decay?       Decay                  

The outstanding was 200,000 . However, after the monthly payments

were made, the outstanding became

150.307568 .

Step 4 : Justifying the answer using the formula Pk = F ( 1 + r/n ) –k

Pk = 1330.60499 ( 1 + 0.07 / 12 ) -360

= 163.9383236

What about the beautiful transitions you’ve been seeing?

Exciting new transitions

They are new too.

3 Part Two

Chemical Acidity

The pH scale measures how acidic or basic a substance is. The pH scale ranges from 0 to 14.A pH of 7 is neutral. The pH scale is Logarithmic and as a result, each whole pH value below 7 is ten times more acidic than the next higher value.

Question 1: a. If an acid has an H + concentration of 0.0001M find the pH. Is it acidic or basic?

Solving Question1 part 2

a.Step1:pH= -log[H+] = -log[0.001] = 3

Step2:pH < 7>> It’s acidic

b.Fruits Antacid tabletpH = -log[H+] pH = -log[H+] 2.2 = -log[H+] 10.1= -log[H+][H+]= 10-2.2 [H+]= 10-2.2

[H+]fruits/[H+]Antacid tablet

10-2.2 /10-2.2 = 79432823.5

Question 2:

Loudness is measured in decibels. The formula for the loudness of a sound is given by “dB =10 log[ I / I0 ]’’ where I0 is the intensity of ’’threshold sound ", or sound that can barely be perceived. Other sounds are defined in terms of how many times more intense they are than threshold sound.

a. Express the formula so that the expression is written as a single logarithm.

dB = 10 (log I – log I0)dB = 10 (log I) –10 (log I0)

b. If a cat’s purr is about 316 times as intense as thresholdSound, then what is the decibel rating of the cat’s purr?

dB = 10 log (I / I0)dB = 10 log (316 I0 / I0)=24.99 ≈ 25 decibels

b. Considering that prolonged exposure to sounds above 85 decibels can cause hearing damage or loss, and considering that a gunshot from a “22- rimfire” rifle has an intensity of about I= (2.5 x 1013) I0should you follow the rules and wear ear protection when relaxing at the rifle range?

dB = 10 log (I / I0)dB = 10 log ((2.5 x 1013) I0 / I0)

=34 decibels

No need to wear ear protection in this case because dB=34 decibels < 85 decibels

Question 3: Many logarithmic expressions may be rewritten either expanded or condensed. Expanding is breaking down a complicated expression into simpler components. Condensing is the reverse of this process. Now why is logaxn = logax ?

logax2 = loga(x.x) = logax + logax = 2logax

Logax3 = loga(x2 . X) = logax2 + logax = 2 logax + logax = 3 logax

Similarly : logaxn = logax

Or logaxn = logaxLet m= logax x=am

Xn = (am)n

logaxn = logaamn = mn

Subsitute for m = logax

logaxn = n logax

Question 4: Show that the sum log1/2 + log2/3 + log3/4 …. + log99/100 equals to a constant. And find the constant.

(log1 – log2) + (log2- log3) + (log3 – log4) + … + (log99 – log100)= log1 – log100= log1/100=log 0.01= -2 constant

End Of PresentationG11.51

Khuloud AlKaabiMaitha Al Mundhari