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Finding the equation of an exponential curve using ln

Finding the equation of an exponential curve using ln. The table shows the amount of caffeine in the bloodstream after a time t mins. Plot the points and verify that the curve is exponential. The equation is N = N 0 e -bt using the exponential function e.

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Finding the equation of an exponential curve using ln

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  1. Finding the equation of an exponential curve using ln The table shows the amount of caffeine in the bloodstream after a time t mins Plot the points and verify that the curve is exponential

  2. The equation is N = N0e-bt using the exponential function e N = Number present after a time period t N0 = Number present initially (time t = 0) t = units of time b = constant

  3. The equation is N = N0ebt Taking lns of both sides ln N = ln(N0 ebt) ln N = ln N0 + lnebt Using the addition rule ln(AB) = lnA + lnB ln N = ln N0 + lne Using the drop down infront rule. ln N = ln N0 + bt lne = 1 ln N = bt + lnN0 Rearranging to match with y=mx + c B A bt

  4. ln N = b t + lnN0 Matching up with y=mx + c : y = ln N gradient = b x = t C = ln N0 So make a new table of values x = t y = lnN

  5. Plot t values on the x axis and lnN values on the y axis

  6. The equation of the line is y = -0.1202x + 4.8292 ln N = b t + lnN0 Matching up : gradient = b = -0.1202 so b = -0.1202 C = ln N0 = 4.8292 To find N0 use forwards and back 4.8292e it = N0 N0ln it = 4.8292 N0 = e4.8282 =125.11

  7. The exponential equation is N = N0ebt N = 125.11×e -0.1202t We do not need to find out how long for the caffeine to ½ .

  8. Using the Equation The exponential curve for the data earlier was given by N = 125.11e -0.1202t. So if the time t is given it is easy to work out the amount of caffeine.  Replace t by the time required If t = 12mins then N = 125.11e -0.120212 = 29.6mg

  9. But if the caffeine N is given then we have to use lns to solve for t. Find the time to reach 50mg N = 125.11e -0.1202t Using lns 50 = 125.11e -0.1202t Forwards t  ×–0.1202  e it  ×125.11 = 50 50 ÷125.11  ln it  ÷–0.1202 Backwards

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