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Introducing Inventor Of Logarithms

Math core project - term3. Introducing Inventor Of Logarithms. The Main PPT Index. 1. 3. 2. Introduction. Part one. Part Two. Click on the one of circles above to go to the slide you need . 1. In troduction. Dividing our work. Getting Started W ith The Project.

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Introducing Inventor Of Logarithms

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  1. Math core project - term3 IntroducingInventor Of Logarithms

  2. TheMain PPT Index 1 3 2 Introduction Part one Part Two Click on the one of circles above to go to the slide you need

  3. 1 Introduction Dividing our work

  4. Getting StartedWith The Project 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.

  5. 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: FYI: “Logarithms”: 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.

  6. 2 Part One Radiocarbon Dating

  7. 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.)

  8. Solving Question1 Equation:   y=a.e-Kt Calculations  to  find  k: 0.5a =ae-K(5600) both (a) will cancel each other -k.5600/-5600=In0.5/-5600use 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

  9. Question 2: The average cost of a home in Tucson is roughly around‭ $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

  10. Step  3:      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 .

  11. Step 4 : Justifying the answer using the formula Pk = F ( 1 + r/n ) –k Pk = 1330.60499 ( 1 + 0.07 / 12 ) -360 = 163.9383236

  12. Exciting new transitions What about the beautiful transitions you’ve been seeing? They are new too.

  13. 3 Part Two Chemical Acidity

  14. 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?

  15. Solving Question1 part 2 a. Step1: pH= -log[H+] = -log[0.001] = 3 Step2: pH < 7 >> It’s acidic b. Fruits Antacid tablet pH = -log[H+] pH = -log[H+] 2.2 = -log[H+] 10.1= -log[H+] [H+]= 10-2.2 [H+]= 10-2.2 [H+]fruits/[H+]Antacidtablet 10-2.2 /10-2.2 = 79432823.5

  16. 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. 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 threshold Sound, 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

  17. 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

  18. 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 = logax Let m= logaxx=am Xn= (am)n logaxn = logaamn = mn Subsitute for m = logax logaxn = n logax

  19. 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

  20. End Of Presentation G11.51 KhuloudAlKaabi Maitha Al Mundhari

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