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
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