4.5 Apply Properties of
Logarithms
p. 259
What are the three properties of logs?
How do you expand a log? Why?
How do you condense a log?
Properties of Logarithms
Use log53≈.683 and log57≈1.209
•log521 =
•log5(3·7)=
•log53 + log57≈
•.683 + 1.209 =
•1.892
Use log53≈.683 and log57≈1.209
• Approximate:
• log549 =
• log572 =
• 2 log57 ≈
• 2(1.209)=
• 2.418
2. log6 40 = log6(8 • 5)
= log 68 + log 65
Write 40 as 8 • 5.
Product property
≈ 1.161 + 0.898
= 2.059
Simplify.
Expanding Logarithms
• You can use the properties to expand logarithms.
3
7
x
• log2
=
y
• log27x3 - log2y =
• log27 + log2x3 – log2y =
• log27 + 3·log2x – log2y
Your turn!
• Expand:
• log 5mn =
• log 5 + log m + log n
• Expand:
• log58x3 =
• log58 + 3·log5x
Condensing Logarithms
• log 6 + 2 log2 – log 3 =
• log 6 + log 22 – log 3 =
• log (6·22) – log 3 =
62
• log
=
3
2
• log 8
Evaluate
using common logarithms and
natural logarithms.
SOLUTION
Using common logarithms:
log 8
0.9031
log 3 8 = log 3
0.4771
1.893
Using natural logarithms:
2.0794
ln 8
log 3 8 = ln 3
1.0986
1.893
• What are the three properties of logs?
Product—expanded add each, Quotient—
expand subtract, Power—expanded goes
in front of log.
• How do you expand a log? Why?
Use “logb” before each addition or
subtraction change. Power property will
bring down exponents so you can solve for
variables.
• How do you condense a log?
Change any addition to multiplication,
subtraction to division and multiplication to
power. Use one “logb”
Sound Intensity
For a sound with intensity I (in
watts per square meter), the
loudness L(I) of the sound (in
decibels) is given by the
function
L(I) =10 log I
I0
whereI0 is the intensity of a barely audible
sound (about 10–12 watts per square meter). An
artist in a recording studio turns up the volume
of a track so that the sound’s intensity
doubles. By how many decibels does the
loudness increase?
SOLUTION
Let I be the original intensity, so that 2I is the
doubled intensity.
Increase in loudness = L(2I) – L(I) Write an expression.
–10log I
= 10log 2I
I
I
0
0
– log I
= 10 log 2I
I0
I0
= 10 log2 + log I – log II
I0
0
= 10 log 2
3.01
Substitute.
Distributive property
Product property
Simplify.
Use a calculator.
ANSWER The loudness increases by about 3 decibels.
4.5 Assignment
page 262, 7-41 odd
Properties of Logarithms
Day 2
• What is the change of base formula?
• What is its purpose?
Your turn!
• Condense:
• log57 + 3·log5t =
• log57t3
• Condense:
• 3log2x – (log24 + log2y)=
3
x
• log2
4y
Change of base formula:
• a, b, and c are positive numbers with b≠1 and c≠1.
Then:
log b a
• logca = log b c
log a
• logca = log c
(base 10)
ln a
• logca = ln c
(base e)
Examples:
• Use the change of base to evaluate:
• log37 =
• (base 10)
•(base e)
• log 7 ≈
• log 3
• 1.771
•ln 7 ≈
•ln 3
•1.771
Use the change-of-base formula to evaluate
the logarithm.
log 5 8
SOLUTION
log 5 8 = log 8
log 5
0.9031
0.6989
1.292
log 8 14
SOLUTION
log
14
log 8 14 =
log 8
1.146
0.9031
1.269
What is the change of base formula?
log b a
log c a 
lobb c
What is its purpose?
Lets you change on base other than
10 or e to common or natural log.
4.5 Assignment Day 2
Page 262,
16- 42 even,
45-59 odd