This document discusses the natural logarithm (ln) and its relationship to the mathematical constant e. It provides examples of how to calculate ln of different values with and without a calculator. It also describes laws of logarithms like ln(ab) = ln(a) + ln(b) and how to solve exponential equations using logarithms by changing the operation to multiplication or division. Examples are given of finding the value of x in equations involving natural logs and the constant e.

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

2. 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 = x is used to denote y = e x
ln
log

3. Calculating
n
The value of (1 + 1/n) approaches e as n gets
bigger and bigger:
(1 + 1/n)
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

4. Look at the graph below - what relationship do the two functions have?
y =e x
y =
x
y =
ln(x)
f(x) =e
x
then
1
f −(x) =
ln(x)

5. Using the natural log - ln
0
e =1
Use a calculator to find:
ln1
=0
ln e
=1
Without using a calculator
find the value of:
3
=3
4
=4
ln e
ln e
1
ln e =
3
3
ln e2
=2
1
ln e =
2
1
ln
e
= -1
1
ln 3 = -3
e
ln e
n
=n

6. The laws of natural logarithms
ln a + ln b = ln ab
a
ln a − ln b = ln
b
b
ln a = b ln a

7. 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 3 x =
log50
Use the power rule
x log3 =
log50
log50
x=
log3
This process can be used with any
base log, even the natural log.
( )
ln 3 x = ln50
x ln3 = ln50
ln50
x=
ln3
x = 3.56
Now try these, answers to 2 d.p.
x
x=
2.79
x
x=
3.32
4 = 48
2 = 10
x=
3.56
x
2.85 = 0.09
x =2.3
−

8. Find x, if ln x =
8
Remember the base of a
natural log is e.
lne x =
8
ex =
20
Find x, if
Take a natural log of both
sides.
x
ln e =
ln20
Rearrange in index form.
loga b = ⇔ ac
c
b=
Use the power rule.
x ln e =
ln20
x =e8
x=
ln20
x=
2980.96
x=
3
Find x in each of the following:
Find x in each of the following:
x
x=
4.61
x
e = 100
ln x = 10
x=
22026
ln x = 4
x=
54.6
e = 3500
x=
8.16
ln x = 0.5
x=
1.65
e x = 0.25
x =1.39
−