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```					The Natural logarithm
and e
IB SL/HL

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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