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CHAPTER 7

CHAPTER 7. Exponential and Logarithmic Functions. Ch 7.1 Exponential Growth and Decay. Population Growth In laboratory experiment the researchers establish a colony of 100 bacteria and monitor its growth. The experimenters discover that the colony triples in population everyday

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CHAPTER 7

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  1. CHAPTER 7 Exponential and Logarithmic Functions

  2. Ch 7.1 Exponential Growth and Decay Population Growth In laboratory experiment the researchers establish a colony of 100 bacteria and monitor its growth. The experimenters discover that the colony triples in population everyday Population P(t), of bacteria in t days P(0) = 100 P(1) = 100.3 P(2) = [100.3].3 P(3) = P(4) = P(5) = The function P(t) = 100(3) t The no. of bacteria present after 8 days= 100(3) 8 = 656, 100 After 36 hours bacteria present 100 (3)1.5= 520 (approx) Graph

  3. 25,000 20,000 15,000 10,000 5000 GraphOf Exponential Growth( in Graph) Population 1 2 3 4 5 Days

  4. Growth or Decay Factors Functions that describe exponential growth or decay can be expressed in the standard form P(t) = Po a t , where Po = P(0) is the initial value of the function and a is the growth or decay factor. • If a> 1, P(t) is increasing, and a = 1 + r, where r represents percent increase • Example P(t) = 100(2)t Increasing 2 is a growth factor • If 0< a < 1, P(t) is decreasing, and a = 1 – r, where r represents percent decrease • Example P(t) = 100( ) t , Decreasing, is a decay factor For bacteria population we have P(t) = 100.3 t Po = 100 and a = 3 Percent Increase Formula A(t) = P(1 + r) t

  5. Comparing Linear Growth and Exponential Growth (pg 426) Linear FunctionExponential function L(t) or E(t) Let consider the two functions L(t) = 5 + 2t and E(t) = 5.2 t E(t) = 5.2 t 50 L(t) = 5 + 2t 0 1 2 3 4 5 t

  6. Ex 7.1, Pg 429 No 2. A population of 24 fruit flies triples every month. How many fruit flies will there be after 6 months? After 3weeks? ( Assume that a month = 4 weeks) • P(t) = P0 at 1st part P(t) = 24(3)t , P0= 24, a = 3, t = 6 months P(6) = 24 (3)6= 17496 2nd part t = 3 weeks = ¾ th months P(3/4) = 24(3) ¾ =54.78= 55 (approx) Graph and table

  7. Graph Table

  8. No 42. Over the week end the Midland Infirmary identifies four cases of Asian flu. Three days later it has treated a total of ten cases a) Flu cases grow linearly L(t) = mt + b Slope = m = L(t) = 2t + 4 b) Flue grows exponentially E(t) = E0 at E0 = 4, E(t) = 4 at 10 = 4 at = at, = a 3 , t = 3 a = = = 1.357 E(t) = 4(1.357)t Graph

  9. Flu grows exponentially Flu cases grow linearly

  10. 7.2 Exponential Functions ( Pg 434) We define an exponential function to be one of the form f(x) = abx , where b > 0 and b = 1, a = 0 If b < 0 , bx will be negative then b is not a real number for some value of x For example b = -3 , bx = (-3) x , f( ½) = ( -3) ½, is an imaginary number If b= 1, f(x) = 1 x = 1 which is constant function Some examples of exponential functions are f(x) = 5x ,   P(t)= 250(1.7)t g(t) = 2.4(0.3) t The constant a is the y-intercept of the graph because f(0) = a.b0= a.1 = a For examples , we find y-intercepts are f(0)= 50 = 1 P(0) = 250(1.7) 0 = 250 G(0) = 2.4(0.3) 0 = 2.4 The positive constant b is called the base of the exponential function

  11. Properties of Exponential Functions (pg 435) f(x) = abx , where b> 0 and b = 1, a = 0 • 1. Domain : All real numbers • 2. Range: All positive numbers • 3. If b> 1, the function is increasing, if 0< b < 1, the function is decreasing

  12. Graphs of Exponential Functions g(x)= (1/2)x f(x)= 2x (3, 8) (-3, 8) ( 0,2) (-2, 1/4) (2, 1/4) ( 0,1) ( 0,1) (-3, 1/8) (3, 1/8) - 5 5 - 5 5

  13. Using Graphing Calculator Pg 437 y = 2x y = 2x + 3 y = 2x+3

  14. Graphical solution of Exponential Equations by Graphing Calculator( Ex- 5, Pg –440) Enter y1 and y2 Zoom 6 Trace

  15. Exponential Regression (Pg 441) STAT ENTER STAT, RIGHT, 0, FOR EXP REG, PRESS ENTER PRESS Y= VARS, 5, RIGHT, RIGHT, ENTER PRESS ZOOM 9

  16. 7.3 Logarithms (Pg 449) Suppose a colony of bacteria doubles in size everyday. If the colony starts with 50 bacteria, how long will it be before there are 800 bacteria ? Example P(x) = 50. 2x ,when P(x) = 800 According to statement 800 = 50.2x Dividing both sides by 50 yields 16 = 2x What power must we raise 2 in order to get 16 ? Because 2 4 = 16 Log2 16 = 4 In other words, we solve an exponential equation by computing a logarithm. Check x = 4 P(4) = 50. 2x = 800

  17. Logarithmic Function ( pg 450 - 451) • y = log b x and x = by For any base b > 0 • log b b= 1 because b1 = b • log b 1= 0 because b0 = 1 • log b b x = x because bx = b x

  18. Steps for Solving Exponential EquationsPg( 454) • Isolate the power on one side of the equation • Rewrite the equation in logarithmic form • Use a calculator, if necessary, to evaluate the logarithm • Solve for the variable

  19. 7.3 No. 40, Pg 458 • The elevation of Mount McKinley, the highest mountain in the United States, is 20,320 feet. What is the atmospheric pressure at the top ? P(a) = 30(10 )-0.9a , Where a= altitude in miles and P = atmospheric pressure in inches of mercury X min = 0 Ymax = 9.4 Xmax = 0 Ymin= 30 A= 20,320 feet= 20,320(1/5280) = 3.8485 miles ( 1mile = 5280 feet) P = 30(10) –(0.09)(3.8485) =13.51inch Check in gr. calculator

  20. 7.4 Logarithmic Functions (pg 461- 462) Logarithmic function Inverse of function

  21. Properties of Logarithmic Functions (Pg 463) • y = log b x and x = by • 1. Domain : All positive real numbers • 2. Range : All real numbers • 3. The graphs of y = log b x and x = by • are symmetric about the line y = x

  22. Evaluating Logarithmic FunctionsUse Log key on a calculatorEx 7.4, Example 2, pg 464 • Let f(x) = log 10 x , Evaluate the following • A) f(35) = log 10 35 = 1.544 • B) f(-8) = , -8 is not the domain of f , f(-8), or log 10 (-8) is undefined • C) 2f(16) + 1 = 2 log 10 16 + 1 • = 2(1.204) + 1 = 3.408 In calculator

  23. Example 2, pg 464 Evaluate the expression log 10 Mf + 1 T = Mo K For k = 0.028, Mf = 1832 and Mo = 15.3 T = log 10 1832 + 1 15.3 = log 10 ( 120.739) 2.082 = 74.35 0.028 0.028 = 0.028 In calculator

  24. Ex 7.4 ,No 12, Pg 469 T = H log 10 , H= 5730, N = 180, N0= 920 log 10 T = 5730 log 10 180 = 13486.33975 920 log 10 ( ) N0 In calulator

  25. 7.6 The Natural Base ( pg 484) • Natural logarithmic function (ln x) In general, y= ln x if and only if ey = x • Example e 2.3 = 10 or ln 10 = 2.3 • In particular ln e = 1 because e 1 = e ln 1 = 0 because e0 = 1 y = e x y = ln x y = x

  26. Properties of Natural Logarithms (pg 485) If x, y > 0, then • ln(xy) = ln x + ln y • ln = ln x – ln y • ln xm = m ln x Useful Properties ln ex = x e lnx = x

  27. Ex 7.6 (Pg 491) No 9. The number of bacteria in a culture grows according to the function N(t) = N0 e 0.04t , N0 is the number of bacteria present at time t = 0 and t is the time in hours. • Growth law N(t) = 6000 e 0.04t c) graph d) After 24 years, there were N(24) = 6000 e 0.04 ( 24) = 15,670 • Let N(t) = 100,000; 100,000 = 6000 e 0.04t DIVIDE BY 6000 AND REDUCE = e 0.04 t Change to logarithmic form : 0.04t = loge = ln t = ln = 70.3 ( divide by 0.04) There will be 100,000 bacteria present after about 70.3 15000 10000 5000 10 20

  28. Ex 7.6, Pg 492 Solve, Round your answer to two decimal places No 22 22.26 = 5.3 e 0.4x 2.7 = e 1.2x ( Divide by 2.3 ) Change to logarithmic form 1.2x = ln 2.7 x = = 0.8277 Solve each equation for the specified variable No. 31 y = k(1- e - t), for t = 1- e – t (Divide by k) e – t = 1 – -t = ln( 1- ) t = - ln ( 1 - ) = ln

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