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Natural Logs

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Natural Logs

  1. 1. The Natural logarithm and e Chapter 8 Section 7 p.525
  2. 2. Learning outcome: you should be able to find the inverse of ln(x), you should be able to solve equations by using the relationship between the natural logarithm and e. Note: y = x is used to denote y = e x ln log
  3. 3. Calculating n The value of (1 + 1/n) approaches e as n gets bigger and bigger: (1 + 1/n) n n 1 2.00000 2 2.25000 5 2.48832 10 2.59374 100 2.70481 1,000 2.71692 10,000 2.71815 100,000 2.71827
  4. 4. Look at the graph below - what relationship do the two functions have? y =e x y = x y = ln(x) f(x) =e x then 1 f −(x) = ln(x)
  5. 5. Using the natural log - ln 0 e =1 Use a calculator to find: ln1 =0 ln e =1 Without using a calculator find the value of: 3 =3 4 =4 ln e ln e 1 ln e = 3 3 ln e2 =2 1 ln e = 2 1 ln e = -1 1 ln 3 = -3 e ln e n =n
  6. 6. The laws of natural logarithms ln a + ln b = ln ab a ln a − ln b = ln b b ln a = b ln a
  7. 7. Finding a missing index using logarithms Find x to 2 decimal places using trial and error. 3x = 50 x= 3.56 Far too complicated ... 3x = 50 Take a log of both sides () log 3 x = log50 Use the power rule x log3 = log50 log50 x= log3 This process can be used with any base log, even the natural log. ( ) ln 3 x = ln50 x ln3 = ln50 ln50 x= ln3 x = 3.56 Now try these, answers to 2 d.p. x x= 2.79 x x= 3.32 4 = 48 2 = 10 x= 3.56 x 2.85 = 0.09 x =2.3 −
  8. 8. Find x, if ln x = 8 Remember the base of a natural log is e. lne x = 8 ex = 20 Find x, if Take a natural log of both sides. x ln e = ln20 Rearrange in index form. loga b = ⇔ ac c b= Use the power rule. x ln e = ln20 x =e8 x= ln20 x= 2980.96 x= 3 Find x in each of the following: Find x in each of the following: x x= 4.61 x e = 100 ln x = 10 x= 22026 ln x = 4 x= 54.6 e = 3500 x= 8.16 ln x = 0.5 x= 1.65 e x = 0.25 x =1.39 −

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