Exponential and Logarithmic Functions
Institute of Mathematics, University of the Philippines Diliman
Lecture 19
(IMath, UPD)
Exponential and Logarithmic Functions
Lec. 19
1 / 29
Outline
1
Real Exponents
2
Exponential Functions
3
Equations involving Exponential Expressions
4
Logarithmic Functions
(IMath, UPD)
Exponential and Logarithmic Functions
Lec. 19
2 / 29
Real Exponents
recall: definition and properties of rational exponents
(IMath, UPD)
Exponential and Logarithmic Functions
Lec. 19
3 / 29
Real Exponents
recall: definition and properties of rational exponents
In particular, if
a >
1
and
p, q
∈
with
p < q
,
(IMath, UPD)
Exponential and Logarithmic Functions
Lec. 19
3 / 29
Real Exponents
recall: definition and properties of rational exponents
In particular, if
a >
1
and
p, q
∈
with
p < q
, then
a
p
< a
q
.
(IMath, UPD)
Exponential and Logarithmic Functions
Lec. 19
3 / 29
Real Exponents
recall: definition and properties of rational exponents
In particular, if
a >
1
and
p, q
∈
with
p < q
, then
a
p
< a
q
.
Ex.
2
3
<
2
4
,
(IMath, UPD)
Exponential and Logarithmic Functions
Lec. 19
3 / 29
Real Exponents
recall: definition and properties of rational exponents
In particular, if
a >
1
and
p, q
∈
with
p < q
, then
a
p
< a
q
.
Ex.
2
3
<
2
4
,
3

1
>
3

2
,
(IMath, UPD)
Exponential and Logarithmic Functions
Lec. 19
3 / 29
Real Exponents
recall: definition and properties of rational exponents
In particular, if
a >
1
and
p, q
∈
with
p < q
, then
a
p
< a
q
.
Ex.
2
3
<
2
4
,
3

1
>
3

2
,
5
1
.
3
<
5
1
.
32
(IMath, UPD)
Exponential and Logarithmic Functions
Lec. 19
3 / 29
Real Exponents
recall: definition and properties of rational exponents
In particular, if
a >
1
and
p, q
∈
with
p < q
, then
a
p
< a
q
.
Ex.
2
3
<
2
4
,
3

1
>
3

2
,
5
1
.
3
<
5
1
.
32
2
√
2
?
(IMath, UPD)
Exponential and Logarithmic Functions
Lec. 19
3 / 29
Real Exponents
recall: definition and properties of rational exponents
In particular, if
a >
1
and
p, q
∈
with
p < q
, then
a
p
< a
q
.
Ex.
2
3
<
2
4
,
3

1
>
3

2
,
5
1
.
3
<
5
1
.
32
2
√
2
?
We want to define
2
√
2
such that the property described above holds.
(IMath, UPD)
Exponential and Logarithmic Functions
Lec. 19
3 / 29
Real Exponents
Note that
√
2
≈
1
.
41421359
...
. If
2
√
2
is to be defined such that properties
of exponents would hold, then:
(IMath, UPD)
Exponential and Logarithmic Functions
Lec. 19
4 / 29
Real Exponents
Note that
√
2
≈
1
.
41421359
...
. If
2
√
2
is to be defined such that properties
of exponents would hold, then:
2
1
<
2
√
2
,
(IMath, UPD)
Exponential and Logarithmic Functions
Lec. 19
4 / 29
Real Exponents
Note that
√
2
≈
1
.
41421359
...
. If
2
√
2
is to be defined such that properties
of exponents would hold, then:
2
1
<
2
√
2
,
2
1
.
4
<
2
√
2
,
(IMath, UPD)
Exponential and Logarithmic Functions
Lec. 19
4 / 29
Real Exponents
Note that
√
2
≈
1
.
41421359
...
. If
2
√
2
is to be defined such that properties
of exponents would hold, then:
2
1
<
2
√
2
,
2
1
.
4
<
2
√
2
,
2
1
.
41
<
2
√
2
,
(IMath, UPD)
Exponential and Logarithmic Functions
Lec. 19
4 / 29
Real Exponents
Note that
√
2
≈
1
.
41421359
...
. If
2
√
2
is to be defined such that properties
of exponents would hold, then:
2
1
<
2
√
2
,
2
1
.
4
<
2
√
2
,
2
1
.
41
<
2
√
2
,
2
1
.
414
<
2
√
2
,
...
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 Spring '11
 AaronJamesPorlante
 Exponential Function, Equations, Exponents, Exponential Functions, Logarithmic Functions, chemical substances, Natural logarithm, Logarithm, Institute of Mathematics