8.5 Properties of Logarithms Goal: Use the properties of logarithms to evaluate, expand and condense logarithmic expressions.
Download ReportTranscript 8.5 Properties of Logarithms Goal: Use the properties of logarithms to evaluate, expand and condense logarithmic expressions.
Slide 1
8.5 Properties of Logarithms
Goal: Use the properties of
logarithms to evaluate, expand and
condense logarithmic expressions.
Slide 2
Warm-up
Simplify:
log 100 log 1000
23 5
log 4 64 log 4 16
3 2 1
log 7 49
2
8 log 3 27
log 7 7
2 2
8 log 3 33
log 7 7 4
83
4
24
Slide 3
Properties of Logarithms
Properties of Logarithms
Let b, m, and n be positive numbers such that b ≠ 1.
Product Property
log b mn log b m log b n
m
log b m log b n
n
Quotient Property
log b
Power Property
log b mn n log b m
Slide 4
Example 1
Use log 5 3 0.683 and log 5 7 1.209 to find the
values of the expression to the nearest th ousandth.
7
log 5
3
log 5 21
log 5 9
log 5 (7)(3)
log5 32
log 5 7 log 5 3
2 log5 3
log5 7 log5 3
1.209 0.683
2(0.683)
1.209 0.683
1.892
1.366
0.526
Slide 5
Example 2
Expand the expression. Assume that all variables are positive.
log6 2 x3
log6 2 log6 x3
log6 2 3 log6 x
5x 4
log7
8y
log7 5x 4 log7 8 y
log7 5 log7 x 4 log7 8 log7 y
log7 5 4 log7 x log7 8 log7 y
Slide 6
Example 3
Condense the expression.
log 6 6 log 6 x 3 log 6 y
log 6 6 x 3 log 6 y
log 6 6 x log 6 y
6x
log 6 3
y
3
3 log 4 5 2 log 4 x log 4 9 2 log 4 y
log 4 53 log 4 x 2 log 4 9 log 4 y 2
log 4 125 x 2 log 4 9 log 4 y 2
125 x 2
log 4
9 y2
Slide 7
Loudness of a Sound
Decibel Level
Example
130
Jackhammer
110
Dance club
90
Lawn mower
70
Television
50
Rainfall
30
Soft whisper
10
Rustling leaf
The loudness L of a sound (in decibels) is
related to the intensity I of the sound (in
watts per square meter) by the equation
L 10 log
I
I0
where I0 is an intensity of 10-12 watts per
square meter, roughly the faintest sound
that can be heard by humans.
An air horn emits sound with an intensity I of 1 watt per square meter. Find its decibel
level.
L 10 log
1
10
12
L 10 log 1012
L 10(12)
L 120 decibels
Slide 8
Loudness of a Sound
I
L 10 log
I0
How much louder is the sound of 5 air horns, compared to just one air horn?
5 1
L5 10 log
10 log
5 1
1012
L5 L1
10
10 log
1
10 log 5 1012 log 1012
12
1012
So 5 air horns would be
approximately 7 decibels
more.
1012
5 1
1
10 log 12 log 12
10
10
10log 5 log 10
L1 10 log
12
1
log 1012
10 log 5
10(.7) 7
Slide 9
Change of Base Formula
Logarithms with any base other than 10 or e can be written in terms of common or
natural logarithms using the change-of-base formula. This allows you to evaluate any
logarithm using a calculator.
Change of Base Formula
If a, b, and c are positive numbers with b ≠ 1 and c ≠ 1, then:
log b a
log c a
log b c
In particular,
log a
log c a
log c
and
ln a
log c a
ln c
Slide 10
Example
Evaluate:
log 6 456
log 456
log 6
2.658964843
0.7781512504
3.417
Slide 11
Assignment
Worksheet 8.5
8.5 Properties of Logarithms
Goal: Use the properties of
logarithms to evaluate, expand and
condense logarithmic expressions.
Slide 2
Warm-up
Simplify:
log 100 log 1000
23 5
log 4 64 log 4 16
3 2 1
log 7 49
2
8 log 3 27
log 7 7
2 2
8 log 3 33
log 7 7 4
83
4
24
Slide 3
Properties of Logarithms
Properties of Logarithms
Let b, m, and n be positive numbers such that b ≠ 1.
Product Property
log b mn log b m log b n
m
log b m log b n
n
Quotient Property
log b
Power Property
log b mn n log b m
Slide 4
Example 1
Use log 5 3 0.683 and log 5 7 1.209 to find the
values of the expression to the nearest th ousandth.
7
log 5
3
log 5 21
log 5 9
log 5 (7)(3)
log5 32
log 5 7 log 5 3
2 log5 3
log5 7 log5 3
1.209 0.683
2(0.683)
1.209 0.683
1.892
1.366
0.526
Slide 5
Example 2
Expand the expression. Assume that all variables are positive.
log6 2 x3
log6 2 log6 x3
log6 2 3 log6 x
5x 4
log7
8y
log7 5x 4 log7 8 y
log7 5 log7 x 4 log7 8 log7 y
log7 5 4 log7 x log7 8 log7 y
Slide 6
Example 3
Condense the expression.
log 6 6 log 6 x 3 log 6 y
log 6 6 x 3 log 6 y
log 6 6 x log 6 y
6x
log 6 3
y
3
3 log 4 5 2 log 4 x log 4 9 2 log 4 y
log 4 53 log 4 x 2 log 4 9 log 4 y 2
log 4 125 x 2 log 4 9 log 4 y 2
125 x 2
log 4
9 y2
Slide 7
Loudness of a Sound
Decibel Level
Example
130
Jackhammer
110
Dance club
90
Lawn mower
70
Television
50
Rainfall
30
Soft whisper
10
Rustling leaf
The loudness L of a sound (in decibels) is
related to the intensity I of the sound (in
watts per square meter) by the equation
L 10 log
I
I0
where I0 is an intensity of 10-12 watts per
square meter, roughly the faintest sound
that can be heard by humans.
An air horn emits sound with an intensity I of 1 watt per square meter. Find its decibel
level.
L 10 log
1
10
12
L 10 log 1012
L 10(12)
L 120 decibels
Slide 8
Loudness of a Sound
I
L 10 log
I0
How much louder is the sound of 5 air horns, compared to just one air horn?
5 1
L5 10 log
10 log
5 1
1012
L5 L1
10
10 log
1
10 log 5 1012 log 1012
12
1012
So 5 air horns would be
approximately 7 decibels
more.
1012
5 1
1
10 log 12 log 12
10
10
10log 5 log 10
L1 10 log
12
1
log 1012
10 log 5
10(.7) 7
Slide 9
Change of Base Formula
Logarithms with any base other than 10 or e can be written in terms of common or
natural logarithms using the change-of-base formula. This allows you to evaluate any
logarithm using a calculator.
Change of Base Formula
If a, b, and c are positive numbers with b ≠ 1 and c ≠ 1, then:
log b a
log c a
log b c
In particular,
log a
log c a
log c
and
ln a
log c a
ln c
Slide 10
Example
Evaluate:
log 6 456
log 456
log 6
2.658964843
0.7781512504
3.417
Slide 11
Assignment
Worksheet 8.5