This defines a new function y log a x known as the

This defines a new function
y = log
a
x
, known as the logarithmic
function with base
a
, which is the exponent to which
a
must be
raised to get x, i.e.,
x = a
y
⇔ y = log
a
x .
Thus, the logarithmic function
y = log
a
x
is the inverse of the
exponential function
y = a
x
.
A logarithm of
a number is an
exponent:
log
b
x
is the
power to which
we must raise
b to get x.
For example,
log
2
8 = 3
Because
10
1

102 Unit 4: Types of Functions
There are two important bases:
 a = 10 gives rise to common logarithms, written simply as log x.
 a = e where e ≈ 2.71828 gives rise to natural logarithms, written
as ln x.
Common and natural logarithms may be evaluated numerically by
pressing either the log or
ln
keys, respectively, on a scientific
calculator.
Properties of the function
f ( x ) = log
a
x
1. The domain of the function is the set of all positive real numbers; the
range is the set of all real numbers.
2. For base a > 1, f(x) is increasing. For 0 < a < 1, f(x) is decreasing.
3. At x = 1, y = 0 independent of the base.
The graphs of the logarithmic functions
y = log
2
x
and
y = log
1
2
x
are
shown in Figure on the right.
These logarithmic functions may be
written equivalently as
x = 2
y
and
x =
(
1
2
)
y
, respectively.
Note that these graphs are reflections of
the graphs of
y = 2
x
and
y = 2
− x
,
respectively, in the line
y = x
.
Examples
1. Evaluate the following:
a)
log
8
64
,
b)
log
3
1
81
,
c)
log
16
2
.
Solution.

a) Let
y = log
8
64
, then
8
y
= 64 = 8
2
and so
y = 2.
b) Let y =
log
3
1
81
,
then
3
y
=
1
81
= 3
− 4
and so
y =− 4.
c) Let y =
log
16
2
, then
16
y
= 2
or
(
2
4
) y
= 2
4 y
= 2
and so 4y = 1 and therefore
y =
1
4
.
2. Solve the following for x:
a)
log
4
x = 3,
b)
log
81
x =
3
4
Solution.
a) x = 4
3
= 64.
b)
x = 81
3
4 =
(
81
1
4
)
3
= 3
3
= 27.
Notation
The notation
log ( x )
is understood to be
log
10
( x )
; for example,
log(7) means
log
10
( 7 )
. The notation
ln ( x )
is understood to be
log
e
( x )
; for example,
ln ( 7 )
means
log
e
( 7 )
. Check on your
calculator that
log 12 = 1.0791812
and
¿ 12 = 2.4849066
.
Examples
1. Solve the equation
ln ( x + 4 ) 2 = 3
for x.
Solution.
2ln ( x + 4 )= 3
10
3

104 Unit 4: Types of Functions
ln ( x + 4 )=
3
2
x + 4 = e
1.5
x + 4 = 4.48169
to 5 decimal places
x = 0.48169
to 5 decimal places
2. Express
log
a
3 + log
a
4 − log
a
6
as a single logarithm.
Solution.
6 = log
a
(
3 × 4
)
− ¿ log
a
6
log
a
3 + log
a
4 − log
a
¿
¿ log
a
(
3 × 4
6
)
¿ log
a
2
3. Find the value of
x
satisfying
log
a
x = 3log
a
2 + log
a
20 − log
a
1.6
Solution.
log
a
x = 3log
a
2 + log
a
20 − log
a
1.6
¿ log
a
2
3
+ log
a
20 − log
a
1.6
¿ lo g
a
(
8 × 20
1.6
)
¿ log
a
100
Therefore
x = 100
.
Unit 4: Types of Functions
4.4 The Exponential and
Logarithmic Functions

Examples
4. Solve the equation
ln ( x + 4 ) 2 = 3
for x.
Solution.
2ln ( x + 4 )= 3
ln ( x + 4 )=
3
2
x + 4 = e
1.5
x + 4 = 4.48169
to 5 decimal places
x = 0.48169
to 5 decimal places
5. Express
log
a
3 + log
a
4 − log
a
6
as a single logarithm.
Solution.
6 = log
a
(
3 × 4
)
− ¿ log
a
6
log
a
3 + log
a
4 − log
a
¿
¿ log
a
(
3 × 4
6
)
¿ log
a
2
6. Find the value of
x
satisfying
log
a
x = 3log
a
2 + log
a
20 − log
a
1.6
Solution.
log
a
x = 3log
a
2 + log
a
20 − log
a
1.6
¿ log
a
2
3
+ log
a
20 − log
a
1.6
¿ lo g
a
(
8 × 20
1.6
)
¿ log
a
100
10
5

106 Unit 4: Types of Functions
Therefore
x = 100
.
Unit 5: Mathematics of Finance
5.0 Simple and Compound
Interest Calculations
Today, businesses and individuals are faced with a bewildering array of
loan facilities and investment opportunities. In this section we explain
how these financial calculations are carried out to enable an informed
choice to be made between the various possibilities available.