# 4.5 Apply Properties of Logarithms

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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