Natural logarithms - Welcome to ibmathscom

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					The Natural logarithm
       and e
         IB SL/HL

       Adrian Sparrow

      www.ibmaths.com
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  ln x is used to denote y  loge x



                         
Look at the graph below - what relationship do the two functions have?

                            y  ex              y x



                                     



                                                               y  ln(x)



                                                        




            f(x)  e   x
                            then      f 1(x)  ln(x)
Using the natural log - ln   Without using a calculator
  0
                             find the value of:
 e 1
                                   3
Use a calculator to find:    ln e       =3
 ln1       =0
                                   4
                             ln e       =4
 ln e      =1
                               3    1
                             ln e =
       2                            3
 ln e      =2
                                1
             1               ln 3 = -3
 ln e      =                   
                               e
             2

    1      = -1              ln e   n
                                        =n
 ln
    
    e
The laws of natural logarithms

     ln a  ln b  ln ab
                      a
     ln a  ln b  ln
                      b
         b
     ln a  b ln a
            Finding a missing index using logarithms
Find x to 2 decimal places using   This process can be used with any
trial and error.                   base log, even the natural log.
 3x  50      x  3.56
                                      
                                   ln 3 x  ln50
Far too complicated ...
                                    x ln3  ln50
 3x  50                                ln50
                                  x
Take a log of both sides                 ln3
                                    x  3.56
    
 log 3x  log50
                                   Now try these, answers to 2 d.p.
Use the power rule            
                                     x
 x log3  log50                    4  48               x  2.79

    log50
 x
     log3
                                     x
                                   2  10               x  3.32
                                                   
x  3.56
                                           x
                                   2.85  0.09          x  2.3
                                           
Find x, if ln x  8                Find x, if e  20x


Remember the base of a             Take a natural log of both
natural log is e.                  sides.
                                    
lne x  8                           lnex  ln20
Rearrange in index form.           Use the power rule.
loga b  c  b  ac                 x lne  ln20
                           
xe  8
                                    x  ln20
x  2980.96                         x 3
                           
                            
Find x in each of the following:   Find x in each of the following:
                                   x
ln x  10      x  22026            e  100             x  4.61


ln x  4       x  54.6              x
                                    e  3500            x  8.16
                                           
     
ln x  0.5     x  1.65              x
                                    e  0.25            x  1.39
                                             

				
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posted:5/8/2010
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