4.5 Apply Properties of Logarithms
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Transcript of 4.5 Apply Properties of Logarithms
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. 6
log 40 =6
log (8 • 5)
= 86
log + 56
log
= 2.059
1.161 0.898+
Write 40 as 8 • 5.Product property
Simplify.
≈
Expanding Logarithms• You can use the properties to expand logarithms.
• log2 =
• log27x3 - log2y =
• log27 + log2x3 – log2y =
• log27 + 3·log2x – log2y
yx37
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 =
• log =
• log 8
326 2
SOLUTION
Evaluate using common logarithms and natural logarithms.
Using common logarithms:
Using natural logarithms:
3log 8 = log 8
log 30.90310.4771 1.893
3log 8 = ln 8
ln 32.07941.0986
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”
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
= logL(I) 10 I0
I
Sound Intensity
0Iwhere is the intensity of a barely audible
sound (about 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?
10–12
Product propertySimplify.
SOLUTIONLet I be the original intensity, so that 2I is the doubled intensity.Increase in loudness = L(2I) – L(I)
= log10 I0
Ilog10 2I0
I –
I0
I2I0
I=10 loglog –
= 210 log log I0
I–log I0
I+
ANSWER The loudness increases by about 3 decibels.
10 log 2=3.01
Write an expression.
Substitute.
Distributive property
Use a calculator.
4.5 Assignment
page 262, 7-41 odd
Properties of LogarithmsDay 2
• What is the change of base formula?• What is its purpose?
Your turn!• Condense:
• log57 + 3·log5t =• log57t3
• Condense:
• 3log2x – (log24 + log2y)=
• log2 yx4
3
Change of base formula:• a, b, and c are positive numbers with b≠1 and c≠1.
Then:
• logca =
• logca = (base 10)
• logca = (base e)
ca
b
b
loglog
ca
loglog
ca
lnln
Examples:
• Use the change of base to evaluate:
• log37 =• (base 10)
• log 7 ≈ • log 3• 1.771
•(base e)•ln 7 ≈ •ln 3•1.771
Use the change-of-base formula to evaluate the logarithm.
5log 8
SOLUTION
5log 8 = log 8
log 50.90310.6989 1.292
8log 14
SOLUTION
8log 14 = log 14
log 81.1460.9031 1.269
clobaa
b
bc
loglog
What is the change of base formula?
What is its purpose?Lets you change on base other than 10 or e to common or natural log.
4.5 Assignment Day 2Page 262, 16- 42 even, 45-59 odd