natural logs

8
The Natural logarithm and e Chapter 8 Section 7 p.525

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Page 1: Natural Logs

The Natural logarithm and e

Chapter 8 Section 7

p.525

Page 2: Natural Logs

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

Page 3: Natural Logs

CalculatingThe value of (1 + 1/n)n approaches e as n

gets bigger and bigger:

n (1 + 1/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

Page 4: Natural Logs

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

Page 5: Natural Logs

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

Page 6: Natural Logs

The laws of natural logarithms

lna lnb lnab

lna lnb lna

blnab b lna

Page 7: Natural Logs

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

Page 8: Natural Logs

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