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: is used to denote

y lnx

y loge x

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

y x

y ex

y ln(x)

f(x) ex

f 1(x) ln(x)then

Using the natural log - ln

e0 1

Use a calculator to find:

ln1

lne

lne2

ln e

ln1

e

=0

=1

=2

=

= -1

1

2

Without using a calculator find the value of:

= 3

= 4

=

= -3

= n

lne3

lne4

ln e3

ln1

e3

lnen

1

3

The laws of natural logarithms

lna lnb lnab

lna lnb lna

blnab b lna

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 3x log50

x log3 log50

Use the power rule

x log50

log3

x 3.56

This process can be used with any base log, even the natural log.

ln 3x ln50

x ln3 ln50

x ln50

ln3x 3.56

Now try these, answers to 2 d.p.

4 x 48

2x 10

2.85x 0.09

x 2.79

x 3.32

x 2.3

Find x, if

lnx 8 Find x, if

ex 20

Remember the base of a natural log is e.

lne x 8

Rearrange in index form.

loga b c b ac

x e8

x 2980.96

lnx 10

lnx 4

lnx 0.5

Find x in each of the following:

x 22026

x 54.6

x 1.65

Take a natural log of both sides.

lnex ln20

Use the power rule.

x lne ln20

x ln20

x 3

Find x in each of the following:

ex 100

ex 3500

ex 0.25

x 4.61

x 8.16

x 1.39