Slide 1

6.5 Properties of Logarithms
In this section, we will study the following
topics:


Using the properties of logarithms to evaluate log
expressions



Using the properties of logarithms to expand or
condense log expressions



Using the change-of-base formula
1

Properties of Logarithmic Functions
To solve various types of problems involving logarithms, we
can make use of the following properties of logarithms.
Properties of Logarithms
1 . lo g a 1  0 sin ce a  1
0

2 . lo g a a  1 sin ce a  a
1

3 . lo g a a  x a n d a
x

lo g a x

 x

(se e h o w th e in v e rse fu n ctio n s "u n d o " o n e a n o th e r)

4 . If lo g a x  lo g a y , th e n x  y

("o n e -to -o n e " p ro p e rty)
2

Properties of Logarithms
The four properties of logarithms listed earlier hold true for
natural logarithms as well.
Properties of Natural Logarithms
1.

ln 1  0 sin ce e  1

2.

ln e  1 sin ce e  e

3.

ln e

4.

If ln x  ln y , th e n x  y

0

1

x

 x and e

ln x

 x

(se e h o w th e in v e rse fu n ctio n s "u n d o " o n e a n o th e r)

3

Examples


Find the exact value of each expression without using a
calculator.
a ) log 2 2

c ) ln e

7

5

b) e

d) 8

ln 12

log 8 3

4

Other Properties of Logarithms
Properties of Logarithms
Let a be a positive real number such that a  1, and let n, u, and v be real numbers.
Base a Logarithms

1. log a  uv   log a u  log a v

2. log a

u

 log a u  log a v

v

3. log a u  n log a u
n

Natural Logarithms

1. ln  uv   ln u  ln v

2. ln

u

 ln u  ln v

v
3. ln u  n ln u
n

5

Properties of Logarithms

WARNING!!!!!!

lo g  x  y   lo g
b

b

x  lo g y
b

6

Properties of Logarithms

ANOTHER WARNING!!!!!!

log

b

M  log N
b

log M

log N
b

b

7

Using the Properties of Logarithms to
Expand Log Expressions
Example:

Use the properties of logs to write the expression as the sum
and/or difference of logarithms (EXPAND).

ln

x y
z

4

3

8

Using the Properties of Logarithms to
Expand Log Expressions
Now it’s your turn! 

Use the properties of logs to expand the following expressions:

a)

lo g

10 z
10

b)

lo g

a

3

6

9

Using the Properties of Logarithms to
Expand Log Expressions

c)

ln

 xy 


z



d ) ln

3

t

10

Using the Properties of Logarithms to
Expand Log Expressions

e ) ln

x

2

y

3

11

Using the Properties of Logarithms to
Condense Log Expressions
Use the properties of logs to write each expressions as a
single logarithm (CONDENSE).

a ) log

5

8  log t
5

b ) 2 ln 7  5 ln x

12

Using the Properties of Logarithms to
Condense Log Expressions
c ) 3 ln x  2 ln y  4 ln z

13

Using the Properties of Logarithms to
Condense Log Expressions
d ) 4  ln z  ln ( z  5)   2 ln ( z  5)

14

Change-of-Base Formula
I mentioned in a previous section that the calculator
only has two types of log keys:
COMMON

LOG (BASE 10)

NATURAL

LOG (BASE e).

It’s true that these two types of logarithms are used
most often, but sometimes we want to evaluate
logarithms with bases other than 10 or e.
15

Change-of-Base Formula

To do this on the calculator, we use a CHANGE OF
BASE FORMULA.

We will convert the logarithm with base a into an
equivalent expression involving common logarithms
or natural logarithms.

16

Change-of-Base Formula
Change-of-Base Formula
Let a, b, and x be positive real numbers such that a  1 and b  1. Then
logax can be converted to a different base using any of the following
formulas.

log a x 

log b x
log b a

log a x 

log 10 x
log 10 a

log a x 

ln x
ln a

17

Change-of-Base Formula Examples
Example*:
Use the change-of-base formula to evaluate log7264
a)

using common logarithms

b)

using natural logarithms.

Solution:
a ) log 7 264 

log 10 264

b ) log 7 264 

ln 264

log 10 7

ln 7



2.42160

 2.8655

0.84510



5.57595

 2.8655

The result is the same
whether you use the
common log or the
natural log.

1.94591
18

Change-of-Base Formula Examples
Example:

Use the change-of-base formula to evaluate:
a) lo g 5 4 1 5

b) log

3

3.45

19



End of Section 6.5

20


Slide 2

6.5 Properties of Logarithms
In this section, we will study the following
topics:


Using the properties of logarithms to evaluate log
expressions



Using the properties of logarithms to expand or
condense log expressions



Using the change-of-base formula
1

Properties of Logarithmic Functions
To solve various types of problems involving logarithms, we
can make use of the following properties of logarithms.
Properties of Logarithms
1 . lo g a 1  0 sin ce a  1
0

2 . lo g a a  1 sin ce a  a
1

3 . lo g a a  x a n d a
x

lo g a x

 x

(se e h o w th e in v e rse fu n ctio n s "u n d o " o n e a n o th e r)

4 . If lo g a x  lo g a y , th e n x  y

("o n e -to -o n e " p ro p e rty)
2

Properties of Logarithms
The four properties of logarithms listed earlier hold true for
natural logarithms as well.
Properties of Natural Logarithms
1.

ln 1  0 sin ce e  1

2.

ln e  1 sin ce e  e

3.

ln e

4.

If ln x  ln y , th e n x  y

0

1

x

 x and e

ln x

 x

(se e h o w th e in v e rse fu n ctio n s "u n d o " o n e a n o th e r)

3

Examples


Find the exact value of each expression without using a
calculator.
a ) log 2 2

c ) ln e

7

5

b) e

d) 8

ln 12

log 8 3

4

Other Properties of Logarithms
Properties of Logarithms
Let a be a positive real number such that a  1, and let n, u, and v be real numbers.
Base a Logarithms

1. log a  uv   log a u  log a v

2. log a

u

 log a u  log a v

v

3. log a u  n log a u
n

Natural Logarithms

1. ln  uv   ln u  ln v

2. ln

u

 ln u  ln v

v
3. ln u  n ln u
n

5

Properties of Logarithms

WARNING!!!!!!

lo g  x  y   lo g
b

b

x  lo g y
b

6

Properties of Logarithms

ANOTHER WARNING!!!!!!

log

b

M  log N
b

log M

log N
b

b

7

Using the Properties of Logarithms to
Expand Log Expressions
Example:

Use the properties of logs to write the expression as the sum
and/or difference of logarithms (EXPAND).

ln

x y
z

4

3

8

Using the Properties of Logarithms to
Expand Log Expressions
Now it’s your turn! 

Use the properties of logs to expand the following expressions:

a)

lo g

10 z
10

b)

lo g

a

3

6

9

Using the Properties of Logarithms to
Expand Log Expressions

c)

ln

 xy 


z



d ) ln

3

t

10

Using the Properties of Logarithms to
Expand Log Expressions

e ) ln

x

2

y

3

11

Using the Properties of Logarithms to
Condense Log Expressions
Use the properties of logs to write each expressions as a
single logarithm (CONDENSE).

a ) log

5

8  log t
5

b ) 2 ln 7  5 ln x

12

Using the Properties of Logarithms to
Condense Log Expressions
c ) 3 ln x  2 ln y  4 ln z

13

Using the Properties of Logarithms to
Condense Log Expressions
d ) 4  ln z  ln ( z  5)   2 ln ( z  5)

14

Change-of-Base Formula
I mentioned in a previous section that the calculator
only has two types of log keys:
COMMON

LOG (BASE 10)

NATURAL

LOG (BASE e).

It’s true that these two types of logarithms are used
most often, but sometimes we want to evaluate
logarithms with bases other than 10 or e.
15

Change-of-Base Formula

To do this on the calculator, we use a CHANGE OF
BASE FORMULA.

We will convert the logarithm with base a into an
equivalent expression involving common logarithms
or natural logarithms.

16

Change-of-Base Formula
Change-of-Base Formula
Let a, b, and x be positive real numbers such that a  1 and b  1. Then
logax can be converted to a different base using any of the following
formulas.

log a x 

log b x
log b a

log a x 

log 10 x
log 10 a

log a x 

ln x
ln a

17

Change-of-Base Formula Examples
Example*:
Use the change-of-base formula to evaluate log7264
a)

using common logarithms

b)

using natural logarithms.

Solution:
a ) log 7 264 

log 10 264

b ) log 7 264 

ln 264

log 10 7

ln 7



2.42160

 2.8655

0.84510



5.57595

 2.8655

The result is the same
whether you use the
common log or the
natural log.

1.94591
18

Change-of-Base Formula Examples
Example:

Use the change-of-base formula to evaluate:
a) lo g 5 4 1 5

b) log

3

3.45

19



End of Section 6.5

20


Slide 3

6.5 Properties of Logarithms
In this section, we will study the following
topics:


Using the properties of logarithms to evaluate log
expressions



Using the properties of logarithms to expand or
condense log expressions



Using the change-of-base formula
1

Properties of Logarithmic Functions
To solve various types of problems involving logarithms, we
can make use of the following properties of logarithms.
Properties of Logarithms
1 . lo g a 1  0 sin ce a  1
0

2 . lo g a a  1 sin ce a  a
1

3 . lo g a a  x a n d a
x

lo g a x

 x

(se e h o w th e in v e rse fu n ctio n s "u n d o " o n e a n o th e r)

4 . If lo g a x  lo g a y , th e n x  y

("o n e -to -o n e " p ro p e rty)
2

Properties of Logarithms
The four properties of logarithms listed earlier hold true for
natural logarithms as well.
Properties of Natural Logarithms
1.

ln 1  0 sin ce e  1

2.

ln e  1 sin ce e  e

3.

ln e

4.

If ln x  ln y , th e n x  y

0

1

x

 x and e

ln x

 x

(se e h o w th e in v e rse fu n ctio n s "u n d o " o n e a n o th e r)

3

Examples


Find the exact value of each expression without using a
calculator.
a ) log 2 2

c ) ln e

7

5

b) e

d) 8

ln 12

log 8 3

4

Other Properties of Logarithms
Properties of Logarithms
Let a be a positive real number such that a  1, and let n, u, and v be real numbers.
Base a Logarithms

1. log a  uv   log a u  log a v

2. log a

u

 log a u  log a v

v

3. log a u  n log a u
n

Natural Logarithms

1. ln  uv   ln u  ln v

2. ln

u

 ln u  ln v

v
3. ln u  n ln u
n

5

Properties of Logarithms

WARNING!!!!!!

lo g  x  y   lo g
b

b

x  lo g y
b

6

Properties of Logarithms

ANOTHER WARNING!!!!!!

log

b

M  log N
b

log M

log N
b

b

7

Using the Properties of Logarithms to
Expand Log Expressions
Example:

Use the properties of logs to write the expression as the sum
and/or difference of logarithms (EXPAND).

ln

x y
z

4

3

8

Using the Properties of Logarithms to
Expand Log Expressions
Now it’s your turn! 

Use the properties of logs to expand the following expressions:

a)

lo g

10 z
10

b)

lo g

a

3

6

9

Using the Properties of Logarithms to
Expand Log Expressions

c)

ln

 xy 


z



d ) ln

3

t

10

Using the Properties of Logarithms to
Expand Log Expressions

e ) ln

x

2

y

3

11

Using the Properties of Logarithms to
Condense Log Expressions
Use the properties of logs to write each expressions as a
single logarithm (CONDENSE).

a ) log

5

8  log t
5

b ) 2 ln 7  5 ln x

12

Using the Properties of Logarithms to
Condense Log Expressions
c ) 3 ln x  2 ln y  4 ln z

13

Using the Properties of Logarithms to
Condense Log Expressions
d ) 4  ln z  ln ( z  5)   2 ln ( z  5)

14

Change-of-Base Formula
I mentioned in a previous section that the calculator
only has two types of log keys:
COMMON

LOG (BASE 10)

NATURAL

LOG (BASE e).

It’s true that these two types of logarithms are used
most often, but sometimes we want to evaluate
logarithms with bases other than 10 or e.
15

Change-of-Base Formula

To do this on the calculator, we use a CHANGE OF
BASE FORMULA.

We will convert the logarithm with base a into an
equivalent expression involving common logarithms
or natural logarithms.

16

Change-of-Base Formula
Change-of-Base Formula
Let a, b, and x be positive real numbers such that a  1 and b  1. Then
logax can be converted to a different base using any of the following
formulas.

log a x 

log b x
log b a

log a x 

log 10 x
log 10 a

log a x 

ln x
ln a

17

Change-of-Base Formula Examples
Example*:
Use the change-of-base formula to evaluate log7264
a)

using common logarithms

b)

using natural logarithms.

Solution:
a ) log 7 264 

log 10 264

b ) log 7 264 

ln 264

log 10 7

ln 7



2.42160

 2.8655

0.84510



5.57595

 2.8655

The result is the same
whether you use the
common log or the
natural log.

1.94591
18

Change-of-Base Formula Examples
Example:

Use the change-of-base formula to evaluate:
a) lo g 5 4 1 5

b) log

3

3.45

19



End of Section 6.5

20


Slide 4

6.5 Properties of Logarithms
In this section, we will study the following
topics:


Using the properties of logarithms to evaluate log
expressions



Using the properties of logarithms to expand or
condense log expressions



Using the change-of-base formula
1

Properties of Logarithmic Functions
To solve various types of problems involving logarithms, we
can make use of the following properties of logarithms.
Properties of Logarithms
1 . lo g a 1  0 sin ce a  1
0

2 . lo g a a  1 sin ce a  a
1

3 . lo g a a  x a n d a
x

lo g a x

 x

(se e h o w th e in v e rse fu n ctio n s "u n d o " o n e a n o th e r)

4 . If lo g a x  lo g a y , th e n x  y

("o n e -to -o n e " p ro p e rty)
2

Properties of Logarithms
The four properties of logarithms listed earlier hold true for
natural logarithms as well.
Properties of Natural Logarithms
1.

ln 1  0 sin ce e  1

2.

ln e  1 sin ce e  e

3.

ln e

4.

If ln x  ln y , th e n x  y

0

1

x

 x and e

ln x

 x

(se e h o w th e in v e rse fu n ctio n s "u n d o " o n e a n o th e r)

3

Examples


Find the exact value of each expression without using a
calculator.
a ) log 2 2

c ) ln e

7

5

b) e

d) 8

ln 12

log 8 3

4

Other Properties of Logarithms
Properties of Logarithms
Let a be a positive real number such that a  1, and let n, u, and v be real numbers.
Base a Logarithms

1. log a  uv   log a u  log a v

2. log a

u

 log a u  log a v

v

3. log a u  n log a u
n

Natural Logarithms

1. ln  uv   ln u  ln v

2. ln

u

 ln u  ln v

v
3. ln u  n ln u
n

5

Properties of Logarithms

WARNING!!!!!!

lo g  x  y   lo g
b

b

x  lo g y
b

6

Properties of Logarithms

ANOTHER WARNING!!!!!!

log

b

M  log N
b

log M

log N
b

b

7

Using the Properties of Logarithms to
Expand Log Expressions
Example:

Use the properties of logs to write the expression as the sum
and/or difference of logarithms (EXPAND).

ln

x y
z

4

3

8

Using the Properties of Logarithms to
Expand Log Expressions
Now it’s your turn! 

Use the properties of logs to expand the following expressions:

a)

lo g

10 z
10

b)

lo g

a

3

6

9

Using the Properties of Logarithms to
Expand Log Expressions

c)

ln

 xy 


z



d ) ln

3

t

10

Using the Properties of Logarithms to
Expand Log Expressions

e ) ln

x

2

y

3

11

Using the Properties of Logarithms to
Condense Log Expressions
Use the properties of logs to write each expressions as a
single logarithm (CONDENSE).

a ) log

5

8  log t
5

b ) 2 ln 7  5 ln x

12

Using the Properties of Logarithms to
Condense Log Expressions
c ) 3 ln x  2 ln y  4 ln z

13

Using the Properties of Logarithms to
Condense Log Expressions
d ) 4  ln z  ln ( z  5)   2 ln ( z  5)

14

Change-of-Base Formula
I mentioned in a previous section that the calculator
only has two types of log keys:
COMMON

LOG (BASE 10)

NATURAL

LOG (BASE e).

It’s true that these two types of logarithms are used
most often, but sometimes we want to evaluate
logarithms with bases other than 10 or e.
15

Change-of-Base Formula

To do this on the calculator, we use a CHANGE OF
BASE FORMULA.

We will convert the logarithm with base a into an
equivalent expression involving common logarithms
or natural logarithms.

16

Change-of-Base Formula
Change-of-Base Formula
Let a, b, and x be positive real numbers such that a  1 and b  1. Then
logax can be converted to a different base using any of the following
formulas.

log a x 

log b x
log b a

log a x 

log 10 x
log 10 a

log a x 

ln x
ln a

17

Change-of-Base Formula Examples
Example*:
Use the change-of-base formula to evaluate log7264
a)

using common logarithms

b)

using natural logarithms.

Solution:
a ) log 7 264 

log 10 264

b ) log 7 264 

ln 264

log 10 7

ln 7



2.42160

 2.8655

0.84510



5.57595

 2.8655

The result is the same
whether you use the
common log or the
natural log.

1.94591
18

Change-of-Base Formula Examples
Example:

Use the change-of-base formula to evaluate:
a) lo g 5 4 1 5

b) log

3

3.45

19



End of Section 6.5

20


Slide 5

6.5 Properties of Logarithms
In this section, we will study the following
topics:


Using the properties of logarithms to evaluate log
expressions



Using the properties of logarithms to expand or
condense log expressions



Using the change-of-base formula
1

Properties of Logarithmic Functions
To solve various types of problems involving logarithms, we
can make use of the following properties of logarithms.
Properties of Logarithms
1 . lo g a 1  0 sin ce a  1
0

2 . lo g a a  1 sin ce a  a
1

3 . lo g a a  x a n d a
x

lo g a x

 x

(se e h o w th e in v e rse fu n ctio n s "u n d o " o n e a n o th e r)

4 . If lo g a x  lo g a y , th e n x  y

("o n e -to -o n e " p ro p e rty)
2

Properties of Logarithms
The four properties of logarithms listed earlier hold true for
natural logarithms as well.
Properties of Natural Logarithms
1.

ln 1  0 sin ce e  1

2.

ln e  1 sin ce e  e

3.

ln e

4.

If ln x  ln y , th e n x  y

0

1

x

 x and e

ln x

 x

(se e h o w th e in v e rse fu n ctio n s "u n d o " o n e a n o th e r)

3

Examples


Find the exact value of each expression without using a
calculator.
a ) log 2 2

c ) ln e

7

5

b) e

d) 8

ln 12

log 8 3

4

Other Properties of Logarithms
Properties of Logarithms
Let a be a positive real number such that a  1, and let n, u, and v be real numbers.
Base a Logarithms

1. log a  uv   log a u  log a v

2. log a

u

 log a u  log a v

v

3. log a u  n log a u
n

Natural Logarithms

1. ln  uv   ln u  ln v

2. ln

u

 ln u  ln v

v
3. ln u  n ln u
n

5

Properties of Logarithms

WARNING!!!!!!

lo g  x  y   lo g
b

b

x  lo g y
b

6

Properties of Logarithms

ANOTHER WARNING!!!!!!

log

b

M  log N
b

log M

log N
b

b

7

Using the Properties of Logarithms to
Expand Log Expressions
Example:

Use the properties of logs to write the expression as the sum
and/or difference of logarithms (EXPAND).

ln

x y
z

4

3

8

Using the Properties of Logarithms to
Expand Log Expressions
Now it’s your turn! 

Use the properties of logs to expand the following expressions:

a)

lo g

10 z
10

b)

lo g

a

3

6

9

Using the Properties of Logarithms to
Expand Log Expressions

c)

ln

 xy 


z



d ) ln

3

t

10

Using the Properties of Logarithms to
Expand Log Expressions

e ) ln

x

2

y

3

11

Using the Properties of Logarithms to
Condense Log Expressions
Use the properties of logs to write each expressions as a
single logarithm (CONDENSE).

a ) log

5

8  log t
5

b ) 2 ln 7  5 ln x

12

Using the Properties of Logarithms to
Condense Log Expressions
c ) 3 ln x  2 ln y  4 ln z

13

Using the Properties of Logarithms to
Condense Log Expressions
d ) 4  ln z  ln ( z  5)   2 ln ( z  5)

14

Change-of-Base Formula
I mentioned in a previous section that the calculator
only has two types of log keys:
COMMON

LOG (BASE 10)

NATURAL

LOG (BASE e).

It’s true that these two types of logarithms are used
most often, but sometimes we want to evaluate
logarithms with bases other than 10 or e.
15

Change-of-Base Formula

To do this on the calculator, we use a CHANGE OF
BASE FORMULA.

We will convert the logarithm with base a into an
equivalent expression involving common logarithms
or natural logarithms.

16

Change-of-Base Formula
Change-of-Base Formula
Let a, b, and x be positive real numbers such that a  1 and b  1. Then
logax can be converted to a different base using any of the following
formulas.

log a x 

log b x
log b a

log a x 

log 10 x
log 10 a

log a x 

ln x
ln a

17

Change-of-Base Formula Examples
Example*:
Use the change-of-base formula to evaluate log7264
a)

using common logarithms

b)

using natural logarithms.

Solution:
a ) log 7 264 

log 10 264

b ) log 7 264 

ln 264

log 10 7

ln 7



2.42160

 2.8655

0.84510



5.57595

 2.8655

The result is the same
whether you use the
common log or the
natural log.

1.94591
18

Change-of-Base Formula Examples
Example:

Use the change-of-base formula to evaluate:
a) lo g 5 4 1 5

b) log

3

3.45

19



End of Section 6.5

20


Slide 6

6.5 Properties of Logarithms
In this section, we will study the following
topics:


Using the properties of logarithms to evaluate log
expressions



Using the properties of logarithms to expand or
condense log expressions



Using the change-of-base formula
1

Properties of Logarithmic Functions
To solve various types of problems involving logarithms, we
can make use of the following properties of logarithms.
Properties of Logarithms
1 . lo g a 1  0 sin ce a  1
0

2 . lo g a a  1 sin ce a  a
1

3 . lo g a a  x a n d a
x

lo g a x

 x

(se e h o w th e in v e rse fu n ctio n s "u n d o " o n e a n o th e r)

4 . If lo g a x  lo g a y , th e n x  y

("o n e -to -o n e " p ro p e rty)
2

Properties of Logarithms
The four properties of logarithms listed earlier hold true for
natural logarithms as well.
Properties of Natural Logarithms
1.

ln 1  0 sin ce e  1

2.

ln e  1 sin ce e  e

3.

ln e

4.

If ln x  ln y , th e n x  y

0

1

x

 x and e

ln x

 x

(se e h o w th e in v e rse fu n ctio n s "u n d o " o n e a n o th e r)

3

Examples


Find the exact value of each expression without using a
calculator.
a ) log 2 2

c ) ln e

7

5

b) e

d) 8

ln 12

log 8 3

4

Other Properties of Logarithms
Properties of Logarithms
Let a be a positive real number such that a  1, and let n, u, and v be real numbers.
Base a Logarithms

1. log a  uv   log a u  log a v

2. log a

u

 log a u  log a v

v

3. log a u  n log a u
n

Natural Logarithms

1. ln  uv   ln u  ln v

2. ln

u

 ln u  ln v

v
3. ln u  n ln u
n

5

Properties of Logarithms

WARNING!!!!!!

lo g  x  y   lo g
b

b

x  lo g y
b

6

Properties of Logarithms

ANOTHER WARNING!!!!!!

log

b

M  log N
b

log M

log N
b

b

7

Using the Properties of Logarithms to
Expand Log Expressions
Example:

Use the properties of logs to write the expression as the sum
and/or difference of logarithms (EXPAND).

ln

x y
z

4

3

8

Using the Properties of Logarithms to
Expand Log Expressions
Now it’s your turn! 

Use the properties of logs to expand the following expressions:

a)

lo g

10 z
10

b)

lo g

a

3

6

9

Using the Properties of Logarithms to
Expand Log Expressions

c)

ln

 xy 


z



d ) ln

3

t

10

Using the Properties of Logarithms to
Expand Log Expressions

e ) ln

x

2

y

3

11

Using the Properties of Logarithms to
Condense Log Expressions
Use the properties of logs to write each expressions as a
single logarithm (CONDENSE).

a ) log

5

8  log t
5

b ) 2 ln 7  5 ln x

12

Using the Properties of Logarithms to
Condense Log Expressions
c ) 3 ln x  2 ln y  4 ln z

13

Using the Properties of Logarithms to
Condense Log Expressions
d ) 4  ln z  ln ( z  5)   2 ln ( z  5)

14

Change-of-Base Formula
I mentioned in a previous section that the calculator
only has two types of log keys:
COMMON

LOG (BASE 10)

NATURAL

LOG (BASE e).

It’s true that these two types of logarithms are used
most often, but sometimes we want to evaluate
logarithms with bases other than 10 or e.
15

Change-of-Base Formula

To do this on the calculator, we use a CHANGE OF
BASE FORMULA.

We will convert the logarithm with base a into an
equivalent expression involving common logarithms
or natural logarithms.

16

Change-of-Base Formula
Change-of-Base Formula
Let a, b, and x be positive real numbers such that a  1 and b  1. Then
logax can be converted to a different base using any of the following
formulas.

log a x 

log b x
log b a

log a x 

log 10 x
log 10 a

log a x 

ln x
ln a

17

Change-of-Base Formula Examples
Example*:
Use the change-of-base formula to evaluate log7264
a)

using common logarithms

b)

using natural logarithms.

Solution:
a ) log 7 264 

log 10 264

b ) log 7 264 

ln 264

log 10 7

ln 7



2.42160

 2.8655

0.84510



5.57595

 2.8655

The result is the same
whether you use the
common log or the
natural log.

1.94591
18

Change-of-Base Formula Examples
Example:

Use the change-of-base formula to evaluate:
a) lo g 5 4 1 5

b) log

3

3.45

19



End of Section 6.5

20


Slide 7

6.5 Properties of Logarithms
In this section, we will study the following
topics:


Using the properties of logarithms to evaluate log
expressions



Using the properties of logarithms to expand or
condense log expressions



Using the change-of-base formula
1

Properties of Logarithmic Functions
To solve various types of problems involving logarithms, we
can make use of the following properties of logarithms.
Properties of Logarithms
1 . lo g a 1  0 sin ce a  1
0

2 . lo g a a  1 sin ce a  a
1

3 . lo g a a  x a n d a
x

lo g a x

 x

(se e h o w th e in v e rse fu n ctio n s "u n d o " o n e a n o th e r)

4 . If lo g a x  lo g a y , th e n x  y

("o n e -to -o n e " p ro p e rty)
2

Properties of Logarithms
The four properties of logarithms listed earlier hold true for
natural logarithms as well.
Properties of Natural Logarithms
1.

ln 1  0 sin ce e  1

2.

ln e  1 sin ce e  e

3.

ln e

4.

If ln x  ln y , th e n x  y

0

1

x

 x and e

ln x

 x

(se e h o w th e in v e rse fu n ctio n s "u n d o " o n e a n o th e r)

3

Examples


Find the exact value of each expression without using a
calculator.
a ) log 2 2

c ) ln e

7

5

b) e

d) 8

ln 12

log 8 3

4

Other Properties of Logarithms
Properties of Logarithms
Let a be a positive real number such that a  1, and let n, u, and v be real numbers.
Base a Logarithms

1. log a  uv   log a u  log a v

2. log a

u

 log a u  log a v

v

3. log a u  n log a u
n

Natural Logarithms

1. ln  uv   ln u  ln v

2. ln

u

 ln u  ln v

v
3. ln u  n ln u
n

5

Properties of Logarithms

WARNING!!!!!!

lo g  x  y   lo g
b

b

x  lo g y
b

6

Properties of Logarithms

ANOTHER WARNING!!!!!!

log

b

M  log N
b

log M

log N
b

b

7

Using the Properties of Logarithms to
Expand Log Expressions
Example:

Use the properties of logs to write the expression as the sum
and/or difference of logarithms (EXPAND).

ln

x y
z

4

3

8

Using the Properties of Logarithms to
Expand Log Expressions
Now it’s your turn! 

Use the properties of logs to expand the following expressions:

a)

lo g

10 z
10

b)

lo g

a

3

6

9

Using the Properties of Logarithms to
Expand Log Expressions

c)

ln

 xy 


z



d ) ln

3

t

10

Using the Properties of Logarithms to
Expand Log Expressions

e ) ln

x

2

y

3

11

Using the Properties of Logarithms to
Condense Log Expressions
Use the properties of logs to write each expressions as a
single logarithm (CONDENSE).

a ) log

5

8  log t
5

b ) 2 ln 7  5 ln x

12

Using the Properties of Logarithms to
Condense Log Expressions
c ) 3 ln x  2 ln y  4 ln z

13

Using the Properties of Logarithms to
Condense Log Expressions
d ) 4  ln z  ln ( z  5)   2 ln ( z  5)

14

Change-of-Base Formula
I mentioned in a previous section that the calculator
only has two types of log keys:
COMMON

LOG (BASE 10)

NATURAL

LOG (BASE e).

It’s true that these two types of logarithms are used
most often, but sometimes we want to evaluate
logarithms with bases other than 10 or e.
15

Change-of-Base Formula

To do this on the calculator, we use a CHANGE OF
BASE FORMULA.

We will convert the logarithm with base a into an
equivalent expression involving common logarithms
or natural logarithms.

16

Change-of-Base Formula
Change-of-Base Formula
Let a, b, and x be positive real numbers such that a  1 and b  1. Then
logax can be converted to a different base using any of the following
formulas.

log a x 

log b x
log b a

log a x 

log 10 x
log 10 a

log a x 

ln x
ln a

17

Change-of-Base Formula Examples
Example*:
Use the change-of-base formula to evaluate log7264
a)

using common logarithms

b)

using natural logarithms.

Solution:
a ) log 7 264 

log 10 264

b ) log 7 264 

ln 264

log 10 7

ln 7



2.42160

 2.8655

0.84510



5.57595

 2.8655

The result is the same
whether you use the
common log or the
natural log.

1.94591
18

Change-of-Base Formula Examples
Example:

Use the change-of-base formula to evaluate:
a) lo g 5 4 1 5

b) log

3

3.45

19



End of Section 6.5

20


Slide 8

6.5 Properties of Logarithms
In this section, we will study the following
topics:


Using the properties of logarithms to evaluate log
expressions



Using the properties of logarithms to expand or
condense log expressions



Using the change-of-base formula
1

Properties of Logarithmic Functions
To solve various types of problems involving logarithms, we
can make use of the following properties of logarithms.
Properties of Logarithms
1 . lo g a 1  0 sin ce a  1
0

2 . lo g a a  1 sin ce a  a
1

3 . lo g a a  x a n d a
x

lo g a x

 x

(se e h o w th e in v e rse fu n ctio n s "u n d o " o n e a n o th e r)

4 . If lo g a x  lo g a y , th e n x  y

("o n e -to -o n e " p ro p e rty)
2

Properties of Logarithms
The four properties of logarithms listed earlier hold true for
natural logarithms as well.
Properties of Natural Logarithms
1.

ln 1  0 sin ce e  1

2.

ln e  1 sin ce e  e

3.

ln e

4.

If ln x  ln y , th e n x  y

0

1

x

 x and e

ln x

 x

(se e h o w th e in v e rse fu n ctio n s "u n d o " o n e a n o th e r)

3

Examples


Find the exact value of each expression without using a
calculator.
a ) log 2 2

c ) ln e

7

5

b) e

d) 8

ln 12

log 8 3

4

Other Properties of Logarithms
Properties of Logarithms
Let a be a positive real number such that a  1, and let n, u, and v be real numbers.
Base a Logarithms

1. log a  uv   log a u  log a v

2. log a

u

 log a u  log a v

v

3. log a u  n log a u
n

Natural Logarithms

1. ln  uv   ln u  ln v

2. ln

u

 ln u  ln v

v
3. ln u  n ln u
n

5

Properties of Logarithms

WARNING!!!!!!

lo g  x  y   lo g
b

b

x  lo g y
b

6

Properties of Logarithms

ANOTHER WARNING!!!!!!

log

b

M  log N
b

log M

log N
b

b

7

Using the Properties of Logarithms to
Expand Log Expressions
Example:

Use the properties of logs to write the expression as the sum
and/or difference of logarithms (EXPAND).

ln

x y
z

4

3

8

Using the Properties of Logarithms to
Expand Log Expressions
Now it’s your turn! 

Use the properties of logs to expand the following expressions:

a)

lo g

10 z
10

b)

lo g

a

3

6

9

Using the Properties of Logarithms to
Expand Log Expressions

c)

ln

 xy 


z



d ) ln

3

t

10

Using the Properties of Logarithms to
Expand Log Expressions

e ) ln

x

2

y

3

11

Using the Properties of Logarithms to
Condense Log Expressions
Use the properties of logs to write each expressions as a
single logarithm (CONDENSE).

a ) log

5

8  log t
5

b ) 2 ln 7  5 ln x

12

Using the Properties of Logarithms to
Condense Log Expressions
c ) 3 ln x  2 ln y  4 ln z

13

Using the Properties of Logarithms to
Condense Log Expressions
d ) 4  ln z  ln ( z  5)   2 ln ( z  5)

14

Change-of-Base Formula
I mentioned in a previous section that the calculator
only has two types of log keys:
COMMON

LOG (BASE 10)

NATURAL

LOG (BASE e).

It’s true that these two types of logarithms are used
most often, but sometimes we want to evaluate
logarithms with bases other than 10 or e.
15

Change-of-Base Formula

To do this on the calculator, we use a CHANGE OF
BASE FORMULA.

We will convert the logarithm with base a into an
equivalent expression involving common logarithms
or natural logarithms.

16

Change-of-Base Formula
Change-of-Base Formula
Let a, b, and x be positive real numbers such that a  1 and b  1. Then
logax can be converted to a different base using any of the following
formulas.

log a x 

log b x
log b a

log a x 

log 10 x
log 10 a

log a x 

ln x
ln a

17

Change-of-Base Formula Examples
Example*:
Use the change-of-base formula to evaluate log7264
a)

using common logarithms

b)

using natural logarithms.

Solution:
a ) log 7 264 

log 10 264

b ) log 7 264 

ln 264

log 10 7

ln 7



2.42160

 2.8655

0.84510



5.57595

 2.8655

The result is the same
whether you use the
common log or the
natural log.

1.94591
18

Change-of-Base Formula Examples
Example:

Use the change-of-base formula to evaluate:
a) lo g 5 4 1 5

b) log

3

3.45

19



End of Section 6.5

20


Slide 9

6.5 Properties of Logarithms
In this section, we will study the following
topics:


Using the properties of logarithms to evaluate log
expressions



Using the properties of logarithms to expand or
condense log expressions



Using the change-of-base formula
1

Properties of Logarithmic Functions
To solve various types of problems involving logarithms, we
can make use of the following properties of logarithms.
Properties of Logarithms
1 . lo g a 1  0 sin ce a  1
0

2 . lo g a a  1 sin ce a  a
1

3 . lo g a a  x a n d a
x

lo g a x

 x

(se e h o w th e in v e rse fu n ctio n s "u n d o " o n e a n o th e r)

4 . If lo g a x  lo g a y , th e n x  y

("o n e -to -o n e " p ro p e rty)
2

Properties of Logarithms
The four properties of logarithms listed earlier hold true for
natural logarithms as well.
Properties of Natural Logarithms
1.

ln 1  0 sin ce e  1

2.

ln e  1 sin ce e  e

3.

ln e

4.

If ln x  ln y , th e n x  y

0

1

x

 x and e

ln x

 x

(se e h o w th e in v e rse fu n ctio n s "u n d o " o n e a n o th e r)

3

Examples


Find the exact value of each expression without using a
calculator.
a ) log 2 2

c ) ln e

7

5

b) e

d) 8

ln 12

log 8 3

4

Other Properties of Logarithms
Properties of Logarithms
Let a be a positive real number such that a  1, and let n, u, and v be real numbers.
Base a Logarithms

1. log a  uv   log a u  log a v

2. log a

u

 log a u  log a v

v

3. log a u  n log a u
n

Natural Logarithms

1. ln  uv   ln u  ln v

2. ln

u

 ln u  ln v

v
3. ln u  n ln u
n

5

Properties of Logarithms

WARNING!!!!!!

lo g  x  y   lo g
b

b

x  lo g y
b

6

Properties of Logarithms

ANOTHER WARNING!!!!!!

log

b

M  log N
b

log M

log N
b

b

7

Using the Properties of Logarithms to
Expand Log Expressions
Example:

Use the properties of logs to write the expression as the sum
and/or difference of logarithms (EXPAND).

ln

x y
z

4

3

8

Using the Properties of Logarithms to
Expand Log Expressions
Now it’s your turn! 

Use the properties of logs to expand the following expressions:

a)

lo g

10 z
10

b)

lo g

a

3

6

9

Using the Properties of Logarithms to
Expand Log Expressions

c)

ln

 xy 


z



d ) ln

3

t

10

Using the Properties of Logarithms to
Expand Log Expressions

e ) ln

x

2

y

3

11

Using the Properties of Logarithms to
Condense Log Expressions
Use the properties of logs to write each expressions as a
single logarithm (CONDENSE).

a ) log

5

8  log t
5

b ) 2 ln 7  5 ln x

12

Using the Properties of Logarithms to
Condense Log Expressions
c ) 3 ln x  2 ln y  4 ln z

13

Using the Properties of Logarithms to
Condense Log Expressions
d ) 4  ln z  ln ( z  5)   2 ln ( z  5)

14

Change-of-Base Formula
I mentioned in a previous section that the calculator
only has two types of log keys:
COMMON

LOG (BASE 10)

NATURAL

LOG (BASE e).

It’s true that these two types of logarithms are used
most often, but sometimes we want to evaluate
logarithms with bases other than 10 or e.
15

Change-of-Base Formula

To do this on the calculator, we use a CHANGE OF
BASE FORMULA.

We will convert the logarithm with base a into an
equivalent expression involving common logarithms
or natural logarithms.

16

Change-of-Base Formula
Change-of-Base Formula
Let a, b, and x be positive real numbers such that a  1 and b  1. Then
logax can be converted to a different base using any of the following
formulas.

log a x 

log b x
log b a

log a x 

log 10 x
log 10 a

log a x 

ln x
ln a

17

Change-of-Base Formula Examples
Example*:
Use the change-of-base formula to evaluate log7264
a)

using common logarithms

b)

using natural logarithms.

Solution:
a ) log 7 264 

log 10 264

b ) log 7 264 

ln 264

log 10 7

ln 7



2.42160

 2.8655

0.84510



5.57595

 2.8655

The result is the same
whether you use the
common log or the
natural log.

1.94591
18

Change-of-Base Formula Examples
Example:

Use the change-of-base formula to evaluate:
a) lo g 5 4 1 5

b) log

3

3.45

19



End of Section 6.5

20


Slide 10

6.5 Properties of Logarithms
In this section, we will study the following
topics:


Using the properties of logarithms to evaluate log
expressions



Using the properties of logarithms to expand or
condense log expressions



Using the change-of-base formula
1

Properties of Logarithmic Functions
To solve various types of problems involving logarithms, we
can make use of the following properties of logarithms.
Properties of Logarithms
1 . lo g a 1  0 sin ce a  1
0

2 . lo g a a  1 sin ce a  a
1

3 . lo g a a  x a n d a
x

lo g a x

 x

(se e h o w th e in v e rse fu n ctio n s "u n d o " o n e a n o th e r)

4 . If lo g a x  lo g a y , th e n x  y

("o n e -to -o n e " p ro p e rty)
2

Properties of Logarithms
The four properties of logarithms listed earlier hold true for
natural logarithms as well.
Properties of Natural Logarithms
1.

ln 1  0 sin ce e  1

2.

ln e  1 sin ce e  e

3.

ln e

4.

If ln x  ln y , th e n x  y

0

1

x

 x and e

ln x

 x

(se e h o w th e in v e rse fu n ctio n s "u n d o " o n e a n o th e r)

3

Examples


Find the exact value of each expression without using a
calculator.
a ) log 2 2

c ) ln e

7

5

b) e

d) 8

ln 12

log 8 3

4

Other Properties of Logarithms
Properties of Logarithms
Let a be a positive real number such that a  1, and let n, u, and v be real numbers.
Base a Logarithms

1. log a  uv   log a u  log a v

2. log a

u

 log a u  log a v

v

3. log a u  n log a u
n

Natural Logarithms

1. ln  uv   ln u  ln v

2. ln

u

 ln u  ln v

v
3. ln u  n ln u
n

5

Properties of Logarithms

WARNING!!!!!!

lo g  x  y   lo g
b

b

x  lo g y
b

6

Properties of Logarithms

ANOTHER WARNING!!!!!!

log

b

M  log N
b

log M

log N
b

b

7

Using the Properties of Logarithms to
Expand Log Expressions
Example:

Use the properties of logs to write the expression as the sum
and/or difference of logarithms (EXPAND).

ln

x y
z

4

3

8

Using the Properties of Logarithms to
Expand Log Expressions
Now it’s your turn! 

Use the properties of logs to expand the following expressions:

a)

lo g

10 z
10

b)

lo g

a

3

6

9

Using the Properties of Logarithms to
Expand Log Expressions

c)

ln

 xy 


z



d ) ln

3

t

10

Using the Properties of Logarithms to
Expand Log Expressions

e ) ln

x

2

y

3

11

Using the Properties of Logarithms to
Condense Log Expressions
Use the properties of logs to write each expressions as a
single logarithm (CONDENSE).

a ) log

5

8  log t
5

b ) 2 ln 7  5 ln x

12

Using the Properties of Logarithms to
Condense Log Expressions
c ) 3 ln x  2 ln y  4 ln z

13

Using the Properties of Logarithms to
Condense Log Expressions
d ) 4  ln z  ln ( z  5)   2 ln ( z  5)

14

Change-of-Base Formula
I mentioned in a previous section that the calculator
only has two types of log keys:
COMMON

LOG (BASE 10)

NATURAL

LOG (BASE e).

It’s true that these two types of logarithms are used
most often, but sometimes we want to evaluate
logarithms with bases other than 10 or e.
15

Change-of-Base Formula

To do this on the calculator, we use a CHANGE OF
BASE FORMULA.

We will convert the logarithm with base a into an
equivalent expression involving common logarithms
or natural logarithms.

16

Change-of-Base Formula
Change-of-Base Formula
Let a, b, and x be positive real numbers such that a  1 and b  1. Then
logax can be converted to a different base using any of the following
formulas.

log a x 

log b x
log b a

log a x 

log 10 x
log 10 a

log a x 

ln x
ln a

17

Change-of-Base Formula Examples
Example*:
Use the change-of-base formula to evaluate log7264
a)

using common logarithms

b)

using natural logarithms.

Solution:
a ) log 7 264 

log 10 264

b ) log 7 264 

ln 264

log 10 7

ln 7



2.42160

 2.8655

0.84510



5.57595

 2.8655

The result is the same
whether you use the
common log or the
natural log.

1.94591
18

Change-of-Base Formula Examples
Example:

Use the change-of-base formula to evaluate:
a) lo g 5 4 1 5

b) log

3

3.45

19



End of Section 6.5

20


Slide 11

6.5 Properties of Logarithms
In this section, we will study the following
topics:


Using the properties of logarithms to evaluate log
expressions



Using the properties of logarithms to expand or
condense log expressions



Using the change-of-base formula
1

Properties of Logarithmic Functions
To solve various types of problems involving logarithms, we
can make use of the following properties of logarithms.
Properties of Logarithms
1 . lo g a 1  0 sin ce a  1
0

2 . lo g a a  1 sin ce a  a
1

3 . lo g a a  x a n d a
x

lo g a x

 x

(se e h o w th e in v e rse fu n ctio n s "u n d o " o n e a n o th e r)

4 . If lo g a x  lo g a y , th e n x  y

("o n e -to -o n e " p ro p e rty)
2

Properties of Logarithms
The four properties of logarithms listed earlier hold true for
natural logarithms as well.
Properties of Natural Logarithms
1.

ln 1  0 sin ce e  1

2.

ln e  1 sin ce e  e

3.

ln e

4.

If ln x  ln y , th e n x  y

0

1

x

 x and e

ln x

 x

(se e h o w th e in v e rse fu n ctio n s "u n d o " o n e a n o th e r)

3

Examples


Find the exact value of each expression without using a
calculator.
a ) log 2 2

c ) ln e

7

5

b) e

d) 8

ln 12

log 8 3

4

Other Properties of Logarithms
Properties of Logarithms
Let a be a positive real number such that a  1, and let n, u, and v be real numbers.
Base a Logarithms

1. log a  uv   log a u  log a v

2. log a

u

 log a u  log a v

v

3. log a u  n log a u
n

Natural Logarithms

1. ln  uv   ln u  ln v

2. ln

u

 ln u  ln v

v
3. ln u  n ln u
n

5

Properties of Logarithms

WARNING!!!!!!

lo g  x  y   lo g
b

b

x  lo g y
b

6

Properties of Logarithms

ANOTHER WARNING!!!!!!

log

b

M  log N
b

log M

log N
b

b

7

Using the Properties of Logarithms to
Expand Log Expressions
Example:

Use the properties of logs to write the expression as the sum
and/or difference of logarithms (EXPAND).

ln

x y
z

4

3

8

Using the Properties of Logarithms to
Expand Log Expressions
Now it’s your turn! 

Use the properties of logs to expand the following expressions:

a)

lo g

10 z
10

b)

lo g

a

3

6

9

Using the Properties of Logarithms to
Expand Log Expressions

c)

ln

 xy 


z



d ) ln

3

t

10

Using the Properties of Logarithms to
Expand Log Expressions

e ) ln

x

2

y

3

11

Using the Properties of Logarithms to
Condense Log Expressions
Use the properties of logs to write each expressions as a
single logarithm (CONDENSE).

a ) log

5

8  log t
5

b ) 2 ln 7  5 ln x

12

Using the Properties of Logarithms to
Condense Log Expressions
c ) 3 ln x  2 ln y  4 ln z

13

Using the Properties of Logarithms to
Condense Log Expressions
d ) 4  ln z  ln ( z  5)   2 ln ( z  5)

14

Change-of-Base Formula
I mentioned in a previous section that the calculator
only has two types of log keys:
COMMON

LOG (BASE 10)

NATURAL

LOG (BASE e).

It’s true that these two types of logarithms are used
most often, but sometimes we want to evaluate
logarithms with bases other than 10 or e.
15

Change-of-Base Formula

To do this on the calculator, we use a CHANGE OF
BASE FORMULA.

We will convert the logarithm with base a into an
equivalent expression involving common logarithms
or natural logarithms.

16

Change-of-Base Formula
Change-of-Base Formula
Let a, b, and x be positive real numbers such that a  1 and b  1. Then
logax can be converted to a different base using any of the following
formulas.

log a x 

log b x
log b a

log a x 

log 10 x
log 10 a

log a x 

ln x
ln a

17

Change-of-Base Formula Examples
Example*:
Use the change-of-base formula to evaluate log7264
a)

using common logarithms

b)

using natural logarithms.

Solution:
a ) log 7 264 

log 10 264

b ) log 7 264 

ln 264

log 10 7

ln 7



2.42160

 2.8655

0.84510



5.57595

 2.8655

The result is the same
whether you use the
common log or the
natural log.

1.94591
18

Change-of-Base Formula Examples
Example:

Use the change-of-base formula to evaluate:
a) lo g 5 4 1 5

b) log

3

3.45

19



End of Section 6.5

20


Slide 12

6.5 Properties of Logarithms
In this section, we will study the following
topics:


Using the properties of logarithms to evaluate log
expressions



Using the properties of logarithms to expand or
condense log expressions



Using the change-of-base formula
1

Properties of Logarithmic Functions
To solve various types of problems involving logarithms, we
can make use of the following properties of logarithms.
Properties of Logarithms
1 . lo g a 1  0 sin ce a  1
0

2 . lo g a a  1 sin ce a  a
1

3 . lo g a a  x a n d a
x

lo g a x

 x

(se e h o w th e in v e rse fu n ctio n s "u n d o " o n e a n o th e r)

4 . If lo g a x  lo g a y , th e n x  y

("o n e -to -o n e " p ro p e rty)
2

Properties of Logarithms
The four properties of logarithms listed earlier hold true for
natural logarithms as well.
Properties of Natural Logarithms
1.

ln 1  0 sin ce e  1

2.

ln e  1 sin ce e  e

3.

ln e

4.

If ln x  ln y , th e n x  y

0

1

x

 x and e

ln x

 x

(se e h o w th e in v e rse fu n ctio n s "u n d o " o n e a n o th e r)

3

Examples


Find the exact value of each expression without using a
calculator.
a ) log 2 2

c ) ln e

7

5

b) e

d) 8

ln 12

log 8 3

4

Other Properties of Logarithms
Properties of Logarithms
Let a be a positive real number such that a  1, and let n, u, and v be real numbers.
Base a Logarithms

1. log a  uv   log a u  log a v

2. log a

u

 log a u  log a v

v

3. log a u  n log a u
n

Natural Logarithms

1. ln  uv   ln u  ln v

2. ln

u

 ln u  ln v

v
3. ln u  n ln u
n

5

Properties of Logarithms

WARNING!!!!!!

lo g  x  y   lo g
b

b

x  lo g y
b

6

Properties of Logarithms

ANOTHER WARNING!!!!!!

log

b

M  log N
b

log M

log N
b

b

7

Using the Properties of Logarithms to
Expand Log Expressions
Example:

Use the properties of logs to write the expression as the sum
and/or difference of logarithms (EXPAND).

ln

x y
z

4

3

8

Using the Properties of Logarithms to
Expand Log Expressions
Now it’s your turn! 

Use the properties of logs to expand the following expressions:

a)

lo g

10 z
10

b)

lo g

a

3

6

9

Using the Properties of Logarithms to
Expand Log Expressions

c)

ln

 xy 


z



d ) ln

3

t

10

Using the Properties of Logarithms to
Expand Log Expressions

e ) ln

x

2

y

3

11

Using the Properties of Logarithms to
Condense Log Expressions
Use the properties of logs to write each expressions as a
single logarithm (CONDENSE).

a ) log

5

8  log t
5

b ) 2 ln 7  5 ln x

12

Using the Properties of Logarithms to
Condense Log Expressions
c ) 3 ln x  2 ln y  4 ln z

13

Using the Properties of Logarithms to
Condense Log Expressions
d ) 4  ln z  ln ( z  5)   2 ln ( z  5)

14

Change-of-Base Formula
I mentioned in a previous section that the calculator
only has two types of log keys:
COMMON

LOG (BASE 10)

NATURAL

LOG (BASE e).

It’s true that these two types of logarithms are used
most often, but sometimes we want to evaluate
logarithms with bases other than 10 or e.
15

Change-of-Base Formula

To do this on the calculator, we use a CHANGE OF
BASE FORMULA.

We will convert the logarithm with base a into an
equivalent expression involving common logarithms
or natural logarithms.

16

Change-of-Base Formula
Change-of-Base Formula
Let a, b, and x be positive real numbers such that a  1 and b  1. Then
logax can be converted to a different base using any of the following
formulas.

log a x 

log b x
log b a

log a x 

log 10 x
log 10 a

log a x 

ln x
ln a

17

Change-of-Base Formula Examples
Example*:
Use the change-of-base formula to evaluate log7264
a)

using common logarithms

b)

using natural logarithms.

Solution:
a ) log 7 264 

log 10 264

b ) log 7 264 

ln 264

log 10 7

ln 7



2.42160

 2.8655

0.84510



5.57595

 2.8655

The result is the same
whether you use the
common log or the
natural log.

1.94591
18

Change-of-Base Formula Examples
Example:

Use the change-of-base formula to evaluate:
a) lo g 5 4 1 5

b) log

3

3.45

19



End of Section 6.5

20


Slide 13

6.5 Properties of Logarithms
In this section, we will study the following
topics:


Using the properties of logarithms to evaluate log
expressions



Using the properties of logarithms to expand or
condense log expressions



Using the change-of-base formula
1

Properties of Logarithmic Functions
To solve various types of problems involving logarithms, we
can make use of the following properties of logarithms.
Properties of Logarithms
1 . lo g a 1  0 sin ce a  1
0

2 . lo g a a  1 sin ce a  a
1

3 . lo g a a  x a n d a
x

lo g a x

 x

(se e h o w th e in v e rse fu n ctio n s "u n d o " o n e a n o th e r)

4 . If lo g a x  lo g a y , th e n x  y

("o n e -to -o n e " p ro p e rty)
2

Properties of Logarithms
The four properties of logarithms listed earlier hold true for
natural logarithms as well.
Properties of Natural Logarithms
1.

ln 1  0 sin ce e  1

2.

ln e  1 sin ce e  e

3.

ln e

4.

If ln x  ln y , th e n x  y

0

1

x

 x and e

ln x

 x

(se e h o w th e in v e rse fu n ctio n s "u n d o " o n e a n o th e r)

3

Examples


Find the exact value of each expression without using a
calculator.
a ) log 2 2

c ) ln e

7

5

b) e

d) 8

ln 12

log 8 3

4

Other Properties of Logarithms
Properties of Logarithms
Let a be a positive real number such that a  1, and let n, u, and v be real numbers.
Base a Logarithms

1. log a  uv   log a u  log a v

2. log a

u

 log a u  log a v

v

3. log a u  n log a u
n

Natural Logarithms

1. ln  uv   ln u  ln v

2. ln

u

 ln u  ln v

v
3. ln u  n ln u
n

5

Properties of Logarithms

WARNING!!!!!!

lo g  x  y   lo g
b

b

x  lo g y
b

6

Properties of Logarithms

ANOTHER WARNING!!!!!!

log

b

M  log N
b

log M

log N
b

b

7

Using the Properties of Logarithms to
Expand Log Expressions
Example:

Use the properties of logs to write the expression as the sum
and/or difference of logarithms (EXPAND).

ln

x y
z

4

3

8

Using the Properties of Logarithms to
Expand Log Expressions
Now it’s your turn! 

Use the properties of logs to expand the following expressions:

a)

lo g

10 z
10

b)

lo g

a

3

6

9

Using the Properties of Logarithms to
Expand Log Expressions

c)

ln

 xy 


z



d ) ln

3

t

10

Using the Properties of Logarithms to
Expand Log Expressions

e ) ln

x

2

y

3

11

Using the Properties of Logarithms to
Condense Log Expressions
Use the properties of logs to write each expressions as a
single logarithm (CONDENSE).

a ) log

5

8  log t
5

b ) 2 ln 7  5 ln x

12

Using the Properties of Logarithms to
Condense Log Expressions
c ) 3 ln x  2 ln y  4 ln z

13

Using the Properties of Logarithms to
Condense Log Expressions
d ) 4  ln z  ln ( z  5)   2 ln ( z  5)

14

Change-of-Base Formula
I mentioned in a previous section that the calculator
only has two types of log keys:
COMMON

LOG (BASE 10)

NATURAL

LOG (BASE e).

It’s true that these two types of logarithms are used
most often, but sometimes we want to evaluate
logarithms with bases other than 10 or e.
15

Change-of-Base Formula

To do this on the calculator, we use a CHANGE OF
BASE FORMULA.

We will convert the logarithm with base a into an
equivalent expression involving common logarithms
or natural logarithms.

16

Change-of-Base Formula
Change-of-Base Formula
Let a, b, and x be positive real numbers such that a  1 and b  1. Then
logax can be converted to a different base using any of the following
formulas.

log a x 

log b x
log b a

log a x 

log 10 x
log 10 a

log a x 

ln x
ln a

17

Change-of-Base Formula Examples
Example*:
Use the change-of-base formula to evaluate log7264
a)

using common logarithms

b)

using natural logarithms.

Solution:
a ) log 7 264 

log 10 264

b ) log 7 264 

ln 264

log 10 7

ln 7



2.42160

 2.8655

0.84510



5.57595

 2.8655

The result is the same
whether you use the
common log or the
natural log.

1.94591
18

Change-of-Base Formula Examples
Example:

Use the change-of-base formula to evaluate:
a) lo g 5 4 1 5

b) log

3

3.45

19



End of Section 6.5

20


Slide 14

6.5 Properties of Logarithms
In this section, we will study the following
topics:


Using the properties of logarithms to evaluate log
expressions



Using the properties of logarithms to expand or
condense log expressions



Using the change-of-base formula
1

Properties of Logarithmic Functions
To solve various types of problems involving logarithms, we
can make use of the following properties of logarithms.
Properties of Logarithms
1 . lo g a 1  0 sin ce a  1
0

2 . lo g a a  1 sin ce a  a
1

3 . lo g a a  x a n d a
x

lo g a x

 x

(se e h o w th e in v e rse fu n ctio n s "u n d o " o n e a n o th e r)

4 . If lo g a x  lo g a y , th e n x  y

("o n e -to -o n e " p ro p e rty)
2

Properties of Logarithms
The four properties of logarithms listed earlier hold true for
natural logarithms as well.
Properties of Natural Logarithms
1.

ln 1  0 sin ce e  1

2.

ln e  1 sin ce e  e

3.

ln e

4.

If ln x  ln y , th e n x  y

0

1

x

 x and e

ln x

 x

(se e h o w th e in v e rse fu n ctio n s "u n d o " o n e a n o th e r)

3

Examples


Find the exact value of each expression without using a
calculator.
a ) log 2 2

c ) ln e

7

5

b) e

d) 8

ln 12

log 8 3

4

Other Properties of Logarithms
Properties of Logarithms
Let a be a positive real number such that a  1, and let n, u, and v be real numbers.
Base a Logarithms

1. log a  uv   log a u  log a v

2. log a

u

 log a u  log a v

v

3. log a u  n log a u
n

Natural Logarithms

1. ln  uv   ln u  ln v

2. ln

u

 ln u  ln v

v
3. ln u  n ln u
n

5

Properties of Logarithms

WARNING!!!!!!

lo g  x  y   lo g
b

b

x  lo g y
b

6

Properties of Logarithms

ANOTHER WARNING!!!!!!

log

b

M  log N
b

log M

log N
b

b

7

Using the Properties of Logarithms to
Expand Log Expressions
Example:

Use the properties of logs to write the expression as the sum
and/or difference of logarithms (EXPAND).

ln

x y
z

4

3

8

Using the Properties of Logarithms to
Expand Log Expressions
Now it’s your turn! 

Use the properties of logs to expand the following expressions:

a)

lo g

10 z
10

b)

lo g

a

3

6

9

Using the Properties of Logarithms to
Expand Log Expressions

c)

ln

 xy 


z



d ) ln

3

t

10

Using the Properties of Logarithms to
Expand Log Expressions

e ) ln

x

2

y

3

11

Using the Properties of Logarithms to
Condense Log Expressions
Use the properties of logs to write each expressions as a
single logarithm (CONDENSE).

a ) log

5

8  log t
5

b ) 2 ln 7  5 ln x

12

Using the Properties of Logarithms to
Condense Log Expressions
c ) 3 ln x  2 ln y  4 ln z

13

Using the Properties of Logarithms to
Condense Log Expressions
d ) 4  ln z  ln ( z  5)   2 ln ( z  5)

14

Change-of-Base Formula
I mentioned in a previous section that the calculator
only has two types of log keys:
COMMON

LOG (BASE 10)

NATURAL

LOG (BASE e).

It’s true that these two types of logarithms are used
most often, but sometimes we want to evaluate
logarithms with bases other than 10 or e.
15

Change-of-Base Formula

To do this on the calculator, we use a CHANGE OF
BASE FORMULA.

We will convert the logarithm with base a into an
equivalent expression involving common logarithms
or natural logarithms.

16

Change-of-Base Formula
Change-of-Base Formula
Let a, b, and x be positive real numbers such that a  1 and b  1. Then
logax can be converted to a different base using any of the following
formulas.

log a x 

log b x
log b a

log a x 

log 10 x
log 10 a

log a x 

ln x
ln a

17

Change-of-Base Formula Examples
Example*:
Use the change-of-base formula to evaluate log7264
a)

using common logarithms

b)

using natural logarithms.

Solution:
a ) log 7 264 

log 10 264

b ) log 7 264 

ln 264

log 10 7

ln 7



2.42160

 2.8655

0.84510



5.57595

 2.8655

The result is the same
whether you use the
common log or the
natural log.

1.94591
18

Change-of-Base Formula Examples
Example:

Use the change-of-base formula to evaluate:
a) lo g 5 4 1 5

b) log

3

3.45

19



End of Section 6.5

20


Slide 15

6.5 Properties of Logarithms
In this section, we will study the following
topics:


Using the properties of logarithms to evaluate log
expressions



Using the properties of logarithms to expand or
condense log expressions



Using the change-of-base formula
1

Properties of Logarithmic Functions
To solve various types of problems involving logarithms, we
can make use of the following properties of logarithms.
Properties of Logarithms
1 . lo g a 1  0 sin ce a  1
0

2 . lo g a a  1 sin ce a  a
1

3 . lo g a a  x a n d a
x

lo g a x

 x

(se e h o w th e in v e rse fu n ctio n s "u n d o " o n e a n o th e r)

4 . If lo g a x  lo g a y , th e n x  y

("o n e -to -o n e " p ro p e rty)
2

Properties of Logarithms
The four properties of logarithms listed earlier hold true for
natural logarithms as well.
Properties of Natural Logarithms
1.

ln 1  0 sin ce e  1

2.

ln e  1 sin ce e  e

3.

ln e

4.

If ln x  ln y , th e n x  y

0

1

x

 x and e

ln x

 x

(se e h o w th e in v e rse fu n ctio n s "u n d o " o n e a n o th e r)

3

Examples


Find the exact value of each expression without using a
calculator.
a ) log 2 2

c ) ln e

7

5

b) e

d) 8

ln 12

log 8 3

4

Other Properties of Logarithms
Properties of Logarithms
Let a be a positive real number such that a  1, and let n, u, and v be real numbers.
Base a Logarithms

1. log a  uv   log a u  log a v

2. log a

u

 log a u  log a v

v

3. log a u  n log a u
n

Natural Logarithms

1. ln  uv   ln u  ln v

2. ln

u

 ln u  ln v

v
3. ln u  n ln u
n

5

Properties of Logarithms

WARNING!!!!!!

lo g  x  y   lo g
b

b

x  lo g y
b

6

Properties of Logarithms

ANOTHER WARNING!!!!!!

log

b

M  log N
b

log M

log N
b

b

7

Using the Properties of Logarithms to
Expand Log Expressions
Example:

Use the properties of logs to write the expression as the sum
and/or difference of logarithms (EXPAND).

ln

x y
z

4

3

8

Using the Properties of Logarithms to
Expand Log Expressions
Now it’s your turn! 

Use the properties of logs to expand the following expressions:

a)

lo g

10 z
10

b)

lo g

a

3

6

9

Using the Properties of Logarithms to
Expand Log Expressions

c)

ln

 xy 


z



d ) ln

3

t

10

Using the Properties of Logarithms to
Expand Log Expressions

e ) ln

x

2

y

3

11

Using the Properties of Logarithms to
Condense Log Expressions
Use the properties of logs to write each expressions as a
single logarithm (CONDENSE).

a ) log

5

8  log t
5

b ) 2 ln 7  5 ln x

12

Using the Properties of Logarithms to
Condense Log Expressions
c ) 3 ln x  2 ln y  4 ln z

13

Using the Properties of Logarithms to
Condense Log Expressions
d ) 4  ln z  ln ( z  5)   2 ln ( z  5)

14

Change-of-Base Formula
I mentioned in a previous section that the calculator
only has two types of log keys:
COMMON

LOG (BASE 10)

NATURAL

LOG (BASE e).

It’s true that these two types of logarithms are used
most often, but sometimes we want to evaluate
logarithms with bases other than 10 or e.
15

Change-of-Base Formula

To do this on the calculator, we use a CHANGE OF
BASE FORMULA.

We will convert the logarithm with base a into an
equivalent expression involving common logarithms
or natural logarithms.

16

Change-of-Base Formula
Change-of-Base Formula
Let a, b, and x be positive real numbers such that a  1 and b  1. Then
logax can be converted to a different base using any of the following
formulas.

log a x 

log b x
log b a

log a x 

log 10 x
log 10 a

log a x 

ln x
ln a

17

Change-of-Base Formula Examples
Example*:
Use the change-of-base formula to evaluate log7264
a)

using common logarithms

b)

using natural logarithms.

Solution:
a ) log 7 264 

log 10 264

b ) log 7 264 

ln 264

log 10 7

ln 7



2.42160

 2.8655

0.84510



5.57595

 2.8655

The result is the same
whether you use the
common log or the
natural log.

1.94591
18

Change-of-Base Formula Examples
Example:

Use the change-of-base formula to evaluate:
a) lo g 5 4 1 5

b) log

3

3.45

19



End of Section 6.5

20


Slide 16

6.5 Properties of Logarithms
In this section, we will study the following
topics:


Using the properties of logarithms to evaluate log
expressions



Using the properties of logarithms to expand or
condense log expressions



Using the change-of-base formula
1

Properties of Logarithmic Functions
To solve various types of problems involving logarithms, we
can make use of the following properties of logarithms.
Properties of Logarithms
1 . lo g a 1  0 sin ce a  1
0

2 . lo g a a  1 sin ce a  a
1

3 . lo g a a  x a n d a
x

lo g a x

 x

(se e h o w th e in v e rse fu n ctio n s "u n d o " o n e a n o th e r)

4 . If lo g a x  lo g a y , th e n x  y

("o n e -to -o n e " p ro p e rty)
2

Properties of Logarithms
The four properties of logarithms listed earlier hold true for
natural logarithms as well.
Properties of Natural Logarithms
1.

ln 1  0 sin ce e  1

2.

ln e  1 sin ce e  e

3.

ln e

4.

If ln x  ln y , th e n x  y

0

1

x

 x and e

ln x

 x

(se e h o w th e in v e rse fu n ctio n s "u n d o " o n e a n o th e r)

3

Examples


Find the exact value of each expression without using a
calculator.
a ) log 2 2

c ) ln e

7

5

b) e

d) 8

ln 12

log 8 3

4

Other Properties of Logarithms
Properties of Logarithms
Let a be a positive real number such that a  1, and let n, u, and v be real numbers.
Base a Logarithms

1. log a  uv   log a u  log a v

2. log a

u

 log a u  log a v

v

3. log a u  n log a u
n

Natural Logarithms

1. ln  uv   ln u  ln v

2. ln

u

 ln u  ln v

v
3. ln u  n ln u
n

5

Properties of Logarithms

WARNING!!!!!!

lo g  x  y   lo g
b

b

x  lo g y
b

6

Properties of Logarithms

ANOTHER WARNING!!!!!!

log

b

M  log N
b

log M

log N
b

b

7

Using the Properties of Logarithms to
Expand Log Expressions
Example:

Use the properties of logs to write the expression as the sum
and/or difference of logarithms (EXPAND).

ln

x y
z

4

3

8

Using the Properties of Logarithms to
Expand Log Expressions
Now it’s your turn! 

Use the properties of logs to expand the following expressions:

a)

lo g

10 z
10

b)

lo g

a

3

6

9

Using the Properties of Logarithms to
Expand Log Expressions

c)

ln

 xy 


z



d ) ln

3

t

10

Using the Properties of Logarithms to
Expand Log Expressions

e ) ln

x

2

y

3

11

Using the Properties of Logarithms to
Condense Log Expressions
Use the properties of logs to write each expressions as a
single logarithm (CONDENSE).

a ) log

5

8  log t
5

b ) 2 ln 7  5 ln x

12

Using the Properties of Logarithms to
Condense Log Expressions
c ) 3 ln x  2 ln y  4 ln z

13

Using the Properties of Logarithms to
Condense Log Expressions
d ) 4  ln z  ln ( z  5)   2 ln ( z  5)

14

Change-of-Base Formula
I mentioned in a previous section that the calculator
only has two types of log keys:
COMMON

LOG (BASE 10)

NATURAL

LOG (BASE e).

It’s true that these two types of logarithms are used
most often, but sometimes we want to evaluate
logarithms with bases other than 10 or e.
15

Change-of-Base Formula

To do this on the calculator, we use a CHANGE OF
BASE FORMULA.

We will convert the logarithm with base a into an
equivalent expression involving common logarithms
or natural logarithms.

16

Change-of-Base Formula
Change-of-Base Formula
Let a, b, and x be positive real numbers such that a  1 and b  1. Then
logax can be converted to a different base using any of the following
formulas.

log a x 

log b x
log b a

log a x 

log 10 x
log 10 a

log a x 

ln x
ln a

17

Change-of-Base Formula Examples
Example*:
Use the change-of-base formula to evaluate log7264
a)

using common logarithms

b)

using natural logarithms.

Solution:
a ) log 7 264 

log 10 264

b ) log 7 264 

ln 264

log 10 7

ln 7



2.42160

 2.8655

0.84510



5.57595

 2.8655

The result is the same
whether you use the
common log or the
natural log.

1.94591
18

Change-of-Base Formula Examples
Example:

Use the change-of-base formula to evaluate:
a) lo g 5 4 1 5

b) log

3

3.45

19



End of Section 6.5

20


Slide 17

6.5 Properties of Logarithms
In this section, we will study the following
topics:


Using the properties of logarithms to evaluate log
expressions



Using the properties of logarithms to expand or
condense log expressions



Using the change-of-base formula
1

Properties of Logarithmic Functions
To solve various types of problems involving logarithms, we
can make use of the following properties of logarithms.
Properties of Logarithms
1 . lo g a 1  0 sin ce a  1
0

2 . lo g a a  1 sin ce a  a
1

3 . lo g a a  x a n d a
x

lo g a x

 x

(se e h o w th e in v e rse fu n ctio n s "u n d o " o n e a n o th e r)

4 . If lo g a x  lo g a y , th e n x  y

("o n e -to -o n e " p ro p e rty)
2

Properties of Logarithms
The four properties of logarithms listed earlier hold true for
natural logarithms as well.
Properties of Natural Logarithms
1.

ln 1  0 sin ce e  1

2.

ln e  1 sin ce e  e

3.

ln e

4.

If ln x  ln y , th e n x  y

0

1

x

 x and e

ln x

 x

(se e h o w th e in v e rse fu n ctio n s "u n d o " o n e a n o th e r)

3

Examples


Find the exact value of each expression without using a
calculator.
a ) log 2 2

c ) ln e

7

5

b) e

d) 8

ln 12

log 8 3

4

Other Properties of Logarithms
Properties of Logarithms
Let a be a positive real number such that a  1, and let n, u, and v be real numbers.
Base a Logarithms

1. log a  uv   log a u  log a v

2. log a

u

 log a u  log a v

v

3. log a u  n log a u
n

Natural Logarithms

1. ln  uv   ln u  ln v

2. ln

u

 ln u  ln v

v
3. ln u  n ln u
n

5

Properties of Logarithms

WARNING!!!!!!

lo g  x  y   lo g
b

b

x  lo g y
b

6

Properties of Logarithms

ANOTHER WARNING!!!!!!

log

b

M  log N
b

log M

log N
b

b

7

Using the Properties of Logarithms to
Expand Log Expressions
Example:

Use the properties of logs to write the expression as the sum
and/or difference of logarithms (EXPAND).

ln

x y
z

4

3

8

Using the Properties of Logarithms to
Expand Log Expressions
Now it’s your turn! 

Use the properties of logs to expand the following expressions:

a)

lo g

10 z
10

b)

lo g

a

3

6

9

Using the Properties of Logarithms to
Expand Log Expressions

c)

ln

 xy 


z



d ) ln

3

t

10

Using the Properties of Logarithms to
Expand Log Expressions

e ) ln

x

2

y

3

11

Using the Properties of Logarithms to
Condense Log Expressions
Use the properties of logs to write each expressions as a
single logarithm (CONDENSE).

a ) log

5

8  log t
5

b ) 2 ln 7  5 ln x

12

Using the Properties of Logarithms to
Condense Log Expressions
c ) 3 ln x  2 ln y  4 ln z

13

Using the Properties of Logarithms to
Condense Log Expressions
d ) 4  ln z  ln ( z  5)   2 ln ( z  5)

14

Change-of-Base Formula
I mentioned in a previous section that the calculator
only has two types of log keys:
COMMON

LOG (BASE 10)

NATURAL

LOG (BASE e).

It’s true that these two types of logarithms are used
most often, but sometimes we want to evaluate
logarithms with bases other than 10 or e.
15

Change-of-Base Formula

To do this on the calculator, we use a CHANGE OF
BASE FORMULA.

We will convert the logarithm with base a into an
equivalent expression involving common logarithms
or natural logarithms.

16

Change-of-Base Formula
Change-of-Base Formula
Let a, b, and x be positive real numbers such that a  1 and b  1. Then
logax can be converted to a different base using any of the following
formulas.

log a x 

log b x
log b a

log a x 

log 10 x
log 10 a

log a x 

ln x
ln a

17

Change-of-Base Formula Examples
Example*:
Use the change-of-base formula to evaluate log7264
a)

using common logarithms

b)

using natural logarithms.

Solution:
a ) log 7 264 

log 10 264

b ) log 7 264 

ln 264

log 10 7

ln 7



2.42160

 2.8655

0.84510



5.57595

 2.8655

The result is the same
whether you use the
common log or the
natural log.

1.94591
18

Change-of-Base Formula Examples
Example:

Use the change-of-base formula to evaluate:
a) lo g 5 4 1 5

b) log

3

3.45

19



End of Section 6.5

20


Slide 18

6.5 Properties of Logarithms
In this section, we will study the following
topics:


Using the properties of logarithms to evaluate log
expressions



Using the properties of logarithms to expand or
condense log expressions



Using the change-of-base formula
1

Properties of Logarithmic Functions
To solve various types of problems involving logarithms, we
can make use of the following properties of logarithms.
Properties of Logarithms
1 . lo g a 1  0 sin ce a  1
0

2 . lo g a a  1 sin ce a  a
1

3 . lo g a a  x a n d a
x

lo g a x

 x

(se e h o w th e in v e rse fu n ctio n s "u n d o " o n e a n o th e r)

4 . If lo g a x  lo g a y , th e n x  y

("o n e -to -o n e " p ro p e rty)
2

Properties of Logarithms
The four properties of logarithms listed earlier hold true for
natural logarithms as well.
Properties of Natural Logarithms
1.

ln 1  0 sin ce e  1

2.

ln e  1 sin ce e  e

3.

ln e

4.

If ln x  ln y , th e n x  y

0

1

x

 x and e

ln x

 x

(se e h o w th e in v e rse fu n ctio n s "u n d o " o n e a n o th e r)

3

Examples


Find the exact value of each expression without using a
calculator.
a ) log 2 2

c ) ln e

7

5

b) e

d) 8

ln 12

log 8 3

4

Other Properties of Logarithms
Properties of Logarithms
Let a be a positive real number such that a  1, and let n, u, and v be real numbers.
Base a Logarithms

1. log a  uv   log a u  log a v

2. log a

u

 log a u  log a v

v

3. log a u  n log a u
n

Natural Logarithms

1. ln  uv   ln u  ln v

2. ln

u

 ln u  ln v

v
3. ln u  n ln u
n

5

Properties of Logarithms

WARNING!!!!!!

lo g  x  y   lo g
b

b

x  lo g y
b

6

Properties of Logarithms

ANOTHER WARNING!!!!!!

log

b

M  log N
b

log M

log N
b

b

7

Using the Properties of Logarithms to
Expand Log Expressions
Example:

Use the properties of logs to write the expression as the sum
and/or difference of logarithms (EXPAND).

ln

x y
z

4

3

8

Using the Properties of Logarithms to
Expand Log Expressions
Now it’s your turn! 

Use the properties of logs to expand the following expressions:

a)

lo g

10 z
10

b)

lo g

a

3

6

9

Using the Properties of Logarithms to
Expand Log Expressions

c)

ln

 xy 


z



d ) ln

3

t

10

Using the Properties of Logarithms to
Expand Log Expressions

e ) ln

x

2

y

3

11

Using the Properties of Logarithms to
Condense Log Expressions
Use the properties of logs to write each expressions as a
single logarithm (CONDENSE).

a ) log

5

8  log t
5

b ) 2 ln 7  5 ln x

12

Using the Properties of Logarithms to
Condense Log Expressions
c ) 3 ln x  2 ln y  4 ln z

13

Using the Properties of Logarithms to
Condense Log Expressions
d ) 4  ln z  ln ( z  5)   2 ln ( z  5)

14

Change-of-Base Formula
I mentioned in a previous section that the calculator
only has two types of log keys:
COMMON

LOG (BASE 10)

NATURAL

LOG (BASE e).

It’s true that these two types of logarithms are used
most often, but sometimes we want to evaluate
logarithms with bases other than 10 or e.
15

Change-of-Base Formula

To do this on the calculator, we use a CHANGE OF
BASE FORMULA.

We will convert the logarithm with base a into an
equivalent expression involving common logarithms
or natural logarithms.

16

Change-of-Base Formula
Change-of-Base Formula
Let a, b, and x be positive real numbers such that a  1 and b  1. Then
logax can be converted to a different base using any of the following
formulas.

log a x 

log b x
log b a

log a x 

log 10 x
log 10 a

log a x 

ln x
ln a

17

Change-of-Base Formula Examples
Example*:
Use the change-of-base formula to evaluate log7264
a)

using common logarithms

b)

using natural logarithms.

Solution:
a ) log 7 264 

log 10 264

b ) log 7 264 

ln 264

log 10 7

ln 7



2.42160

 2.8655

0.84510



5.57595

 2.8655

The result is the same
whether you use the
common log or the
natural log.

1.94591
18

Change-of-Base Formula Examples
Example:

Use the change-of-base formula to evaluate:
a) lo g 5 4 1 5

b) log

3

3.45

19



End of Section 6.5

20


Slide 19

6.5 Properties of Logarithms
In this section, we will study the following
topics:


Using the properties of logarithms to evaluate log
expressions



Using the properties of logarithms to expand or
condense log expressions



Using the change-of-base formula
1

Properties of Logarithmic Functions
To solve various types of problems involving logarithms, we
can make use of the following properties of logarithms.
Properties of Logarithms
1 . lo g a 1  0 sin ce a  1
0

2 . lo g a a  1 sin ce a  a
1

3 . lo g a a  x a n d a
x

lo g a x

 x

(se e h o w th e in v e rse fu n ctio n s "u n d o " o n e a n o th e r)

4 . If lo g a x  lo g a y , th e n x  y

("o n e -to -o n e " p ro p e rty)
2

Properties of Logarithms
The four properties of logarithms listed earlier hold true for
natural logarithms as well.
Properties of Natural Logarithms
1.

ln 1  0 sin ce e  1

2.

ln e  1 sin ce e  e

3.

ln e

4.

If ln x  ln y , th e n x  y

0

1

x

 x and e

ln x

 x

(se e h o w th e in v e rse fu n ctio n s "u n d o " o n e a n o th e r)

3

Examples


Find the exact value of each expression without using a
calculator.
a ) log 2 2

c ) ln e

7

5

b) e

d) 8

ln 12

log 8 3

4

Other Properties of Logarithms
Properties of Logarithms
Let a be a positive real number such that a  1, and let n, u, and v be real numbers.
Base a Logarithms

1. log a  uv   log a u  log a v

2. log a

u

 log a u  log a v

v

3. log a u  n log a u
n

Natural Logarithms

1. ln  uv   ln u  ln v

2. ln

u

 ln u  ln v

v
3. ln u  n ln u
n

5

Properties of Logarithms

WARNING!!!!!!

lo g  x  y   lo g
b

b

x  lo g y
b

6

Properties of Logarithms

ANOTHER WARNING!!!!!!

log

b

M  log N
b

log M

log N
b

b

7

Using the Properties of Logarithms to
Expand Log Expressions
Example:

Use the properties of logs to write the expression as the sum
and/or difference of logarithms (EXPAND).

ln

x y
z

4

3

8

Using the Properties of Logarithms to
Expand Log Expressions
Now it’s your turn! 

Use the properties of logs to expand the following expressions:

a)

lo g

10 z
10

b)

lo g

a

3

6

9

Using the Properties of Logarithms to
Expand Log Expressions

c)

ln

 xy 


z



d ) ln

3

t

10

Using the Properties of Logarithms to
Expand Log Expressions

e ) ln

x

2

y

3

11

Using the Properties of Logarithms to
Condense Log Expressions
Use the properties of logs to write each expressions as a
single logarithm (CONDENSE).

a ) log

5

8  log t
5

b ) 2 ln 7  5 ln x

12

Using the Properties of Logarithms to
Condense Log Expressions
c ) 3 ln x  2 ln y  4 ln z

13

Using the Properties of Logarithms to
Condense Log Expressions
d ) 4  ln z  ln ( z  5)   2 ln ( z  5)

14

Change-of-Base Formula
I mentioned in a previous section that the calculator
only has two types of log keys:
COMMON

LOG (BASE 10)

NATURAL

LOG (BASE e).

It’s true that these two types of logarithms are used
most often, but sometimes we want to evaluate
logarithms with bases other than 10 or e.
15

Change-of-Base Formula

To do this on the calculator, we use a CHANGE OF
BASE FORMULA.

We will convert the logarithm with base a into an
equivalent expression involving common logarithms
or natural logarithms.

16

Change-of-Base Formula
Change-of-Base Formula
Let a, b, and x be positive real numbers such that a  1 and b  1. Then
logax can be converted to a different base using any of the following
formulas.

log a x 

log b x
log b a

log a x 

log 10 x
log 10 a

log a x 

ln x
ln a

17

Change-of-Base Formula Examples
Example*:
Use the change-of-base formula to evaluate log7264
a)

using common logarithms

b)

using natural logarithms.

Solution:
a ) log 7 264 

log 10 264

b ) log 7 264 

ln 264

log 10 7

ln 7



2.42160

 2.8655

0.84510



5.57595

 2.8655

The result is the same
whether you use the
common log or the
natural log.

1.94591
18

Change-of-Base Formula Examples
Example:

Use the change-of-base formula to evaluate:
a) lo g 5 4 1 5

b) log

3

3.45

19



End of Section 6.5

20


Slide 20

6.5 Properties of Logarithms
In this section, we will study the following
topics:


Using the properties of logarithms to evaluate log
expressions



Using the properties of logarithms to expand or
condense log expressions



Using the change-of-base formula
1

Properties of Logarithmic Functions
To solve various types of problems involving logarithms, we
can make use of the following properties of logarithms.
Properties of Logarithms
1 . lo g a 1  0 sin ce a  1
0

2 . lo g a a  1 sin ce a  a
1

3 . lo g a a  x a n d a
x

lo g a x

 x

(se e h o w th e in v e rse fu n ctio n s "u n d o " o n e a n o th e r)

4 . If lo g a x  lo g a y , th e n x  y

("o n e -to -o n e " p ro p e rty)
2

Properties of Logarithms
The four properties of logarithms listed earlier hold true for
natural logarithms as well.
Properties of Natural Logarithms
1.

ln 1  0 sin ce e  1

2.

ln e  1 sin ce e  e

3.

ln e

4.

If ln x  ln y , th e n x  y

0

1

x

 x and e

ln x

 x

(se e h o w th e in v e rse fu n ctio n s "u n d o " o n e a n o th e r)

3

Examples


Find the exact value of each expression without using a
calculator.
a ) log 2 2

c ) ln e

7

5

b) e

d) 8

ln 12

log 8 3

4

Other Properties of Logarithms
Properties of Logarithms
Let a be a positive real number such that a  1, and let n, u, and v be real numbers.
Base a Logarithms

1. log a  uv   log a u  log a v

2. log a

u

 log a u  log a v

v

3. log a u  n log a u
n

Natural Logarithms

1. ln  uv   ln u  ln v

2. ln

u

 ln u  ln v

v
3. ln u  n ln u
n

5

Properties of Logarithms

WARNING!!!!!!

lo g  x  y   lo g
b

b

x  lo g y
b

6

Properties of Logarithms

ANOTHER WARNING!!!!!!

log

b

M  log N
b

log M

log N
b

b

7

Using the Properties of Logarithms to
Expand Log Expressions
Example:

Use the properties of logs to write the expression as the sum
and/or difference of logarithms (EXPAND).

ln

x y
z

4

3

8

Using the Properties of Logarithms to
Expand Log Expressions
Now it’s your turn! 

Use the properties of logs to expand the following expressions:

a)

lo g

10 z
10

b)

lo g

a

3

6

9

Using the Properties of Logarithms to
Expand Log Expressions

c)

ln

 xy 


z



d ) ln

3

t

10

Using the Properties of Logarithms to
Expand Log Expressions

e ) ln

x

2

y

3

11

Using the Properties of Logarithms to
Condense Log Expressions
Use the properties of logs to write each expressions as a
single logarithm (CONDENSE).

a ) log

5

8  log t
5

b ) 2 ln 7  5 ln x

12

Using the Properties of Logarithms to
Condense Log Expressions
c ) 3 ln x  2 ln y  4 ln z

13

Using the Properties of Logarithms to
Condense Log Expressions
d ) 4  ln z  ln ( z  5)   2 ln ( z  5)

14

Change-of-Base Formula
I mentioned in a previous section that the calculator
only has two types of log keys:
COMMON

LOG (BASE 10)

NATURAL

LOG (BASE e).

It’s true that these two types of logarithms are used
most often, but sometimes we want to evaluate
logarithms with bases other than 10 or e.
15

Change-of-Base Formula

To do this on the calculator, we use a CHANGE OF
BASE FORMULA.

We will convert the logarithm with base a into an
equivalent expression involving common logarithms
or natural logarithms.

16

Change-of-Base Formula
Change-of-Base Formula
Let a, b, and x be positive real numbers such that a  1 and b  1. Then
logax can be converted to a different base using any of the following
formulas.

log a x 

log b x
log b a

log a x 

log 10 x
log 10 a

log a x 

ln x
ln a

17

Change-of-Base Formula Examples
Example*:
Use the change-of-base formula to evaluate log7264
a)

using common logarithms

b)

using natural logarithms.

Solution:
a ) log 7 264 

log 10 264

b ) log 7 264 

ln 264

log 10 7

ln 7



2.42160

 2.8655

0.84510



5.57595

 2.8655

The result is the same
whether you use the
common log or the
natural log.

1.94591
18

Change-of-Base Formula Examples
Example:

Use the change-of-base formula to evaluate:
a) lo g 5 4 1 5

b) log

3

3.45

19



End of Section 6.5

20