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

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

A B Taking lns of both sides ln N = ln(N 0 e bt ) ln N = ln N 0 + lne bt Using the addition rule ln(AB) = lnA + lnB ln N = ln N 0 + lne Using the drop down infront rule. ln N = ln N 0 + bt lne = 1 ln N = bt + lnN 0 Rearranging to match with y=mx + c The equation is N = N 0 e bt bt

So make a new table of values x = t y = lnN ln N = b t + lnN 0 Matching up with y=mx + c : y = ln N gradient = b x = t C = ln N 0

Plot t values on the x axis and lnN values on the y axis tCaffeine Nln N

y = x The equation of the line is ln N = b t + lnN 0 gradient = b = so b = Matching up : C = ln N 0 = To find N 0 use forwards and back N 0 = e = N 0  ln it =  e it = N 0

The exponential equation is N = N 0 e bt N = ×e t We do not need to find out how long for the caffeine to ½.

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

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