Quiz 7-4:

Convert to Convert to exponential formexponential form

1.1.

2.2.

x27log3x1log9

366 x

Convert to Convert to logarithm formlogarithm form

x523.3.

4.4.

8log25.5.

6.6.

Simplify:Simplify:

2

1log2

Find the Inverse:Find the Inverse:

7.7.

8.8.

xy 10)3ln( xy

7-5Properties of Logarithmic Functions

What you’ll learn about

• Properties of Logarithms• Change of Base (review)• Graphs of Logarithmic Functions with Base b• Re-expressing Data

… and whyLogarithms are used extensively in science. You

need to understand their many special properties, so that you can solve problems in the real world.

Properties of Logarithms:

SRRS bbb loglog)(log

4log8log32log 222

235 2

23

25

2 2log2log2log

This is the This is the same thingsame thing as saying: “When you multiply as saying: “When you multiply “ “like” bases of powers, you just add the exponents.”like” bases of powers, you just add the exponents.”

Logarithmic version of the Exponent property:Logarithmic version of the Exponent property: Product of Powers.Product of Powers.

Product RuleProduct Rule::

?32 xxnmnm xxx

1. 1.

?45log3 5log9log 33 Take your pickTake your pick

Your Turn: “Expand the Product”Your Turn: “Expand the Product”

2. 2.

?18log4

?24log2

3log15log 33 5log3log3log 333

5*3*345““Expanding the Product”Expanding the Product”

)5*7(log2

Your Turn: “Condense the Product”Your Turn: “Condense the Product”

3. 3. ?7log2log 55

4. 4. ?4log9log 33

5log7log 22 35log2

““Condense the Product”Condense the Product”

Properties of Logarithms Quotient RuleQuotient Rule

SRS

Rbbb logloglog

?2

5log3

2log5log 33 ““expand the quotient”expand the quotient”

3log8log ““condense the quotient”condense the quotient”

3

8log

252

5x

x

xLogarithmLogarithm: another way of writing the : another way of writing the exponentexponent

Your Turn:

?16log8log 55

5.5.

6. 6.

5

4log

Condense the quotientCondense the quotient

2log5log 44

Expand the QuotientExpand the Quotient

7

3ln

8. 8.

7. 7.

Properties of Logarithms

10log2 5

52log

Power RulePower Rule

cb Rlog

?10log 25

?32log 2log5

cc Rc blog

25

27log2 2

3

2 5

3log 2

23

2 5log3log

5log23log3 22

Combination of expanding & simplifyingCombination of expanding & simplifying

Your Turn:

9. 9. ?ln3

2

y

x

10. 10. 33

4

logy

x

11. 11. 7

5

4log2m

w

Think of it like this:Think of it like this:

7

5

4log2m

w

Expanding the Logarithm of a Product

Use the properties of logarithms to write the Use the properties of logarithms to write the following as a sum of logarithms or multiple logarithms.following as a sum of logarithms or multiple logarithms.

?)16log( 5 xy 5loglog16log yx

yx log5log2log 4

yx log5log2log4

?3

2log

5

y

x)log53(log2log yx

Your Turn:

Expand these logarithms:Expand these logarithms:

42

25log

y

xyx 436log12.12.

13.13.

Condensing a Logarithmic Expression

Write the following expression as a single logarithm:Write the following expression as a single logarithm:

zyx ln)lnln3( z

yx3ln

Your Turn:

Condense the logarithm:Condense the logarithm:

xlog4)3log22log3( 14.14.

Change-of-Base Formula for Logarithms

Change of Base FormulaChange of Base Formula: : c

aa

b

bc log

loglog

This is This is most oftenmost often used to convert from used to convert from other base logs to log other base logs to log base 10base 10 (since that is (since that is what you calculator has on it.) what you calculator has on it.)

5log4

Convert to base 10.Convert to base 10.

4log

5log

10

106021.0

699.0 161.1

Another Example:

Calculate the following using the Calculate the following using the base conversion formula.base conversion formula.

?9log2 2log

9log

301.0

9542.0 17.3

Your turn:

Calculate the following using the Calculate the following using the base conversion formula.base conversion formula.

15.15.

16.16.

?7log8

?9log3

Approximating Expressions

If you don’t have your calculator, you can use givenIf you don’t have your calculator, you can use given values and properties of logarithms to find other logarithms:values and properties of logarithms to find other logarithms:

47712.03log 77815.06log Use these to find:Use these to find: ?2log

3

6log2log

3log6log 47712.77815.0 30103.0

Use your calculator to find: log 2 = ? Use your calculator to find: log 2 = ?

Approximating Expressions

If you don’t have your calculator, you can use givenIf you don’t have your calculator, you can use given values and properties of logarithms to find other logarithms:values and properties of logarithms to find other logarithms:

47712.03log 77815.06log

Use these to find:Use these to find: ?21log

6

3log 6log3log

77815.047712.0 30103.0Use your calculator to find: log 1/2 = ? Use your calculator to find: log 1/2 = ?

Approximating Expressions

If you don’t have your calculator, you can use givenIf you don’t have your calculator, you can use given values and properties of logarithms to find other logarithms:values and properties of logarithms to find other logarithms:

47712.03log 77815.06log Use these to find:Use these to find: ?18log )3*6log(18log

3log6log 47712.77815.0 25527.1

Use your calculator to find: log 18 = ? Use your calculator to find: log 18 = ?

Earthquake Intensity

Is measured by the amplitude of the vibrationIs measured by the amplitude of the vibration felt at the measuring station.felt at the measuring station.

BamplitudeR )log(

B: a “fudge factor” to account for weakeningB: a “fudge factor” to account for weakening of the seismic wave from origin to pt of of the seismic wave from origin to pt of measurementmeasurement

Amplitude is measured in Amplitude is measured in metersm 610

Earthquake Magnitude

pH

In chemistry, the acidity of a water-based solution is measured by the concentration of hydrogen ions in the solution (in moles per liter). The hydrogen-ion concentration is written [H+]. The measure of acidity used is pH, the negative of the common log of the hydrogen-ion concentration:

pH = -log [H+]

More acidic solutions have higher hydrogen-ion concentrations and lower pH values.

Newton’s Law of Cooling

A high temperature item will cool off in a lower temperature A high temperature item will cool off in a lower temperature medium in which it is placed. This cooling off process can be medium in which it is placed. This cooling off process can be modeled by the following equation.modeled by the following equation.

ktmom eTTTtT )()(

Temperature Temperature (as a function of time)(as a function of time)

Temperature ofTemperature of the mediumthe medium

Initial TempInitial Temp of the objectof the object

Constant, determinedConstant, determined by the heat transferby the heat transfer characteristics of the materialcharacteristics of the material

TimeTime

Example Newton’s Law of CoolingA hard-boiled egg at temperature 100ºC is placed in 15ºC water to cool. Five minutes later the temperature of the

egg is 55ºC. K = 0.15 When will the egg be 25ºC? kt

mom eTTTtT )()( te 15.0)15100(1525

““isolate the power”isolate the power” te 15.0

)15100(

1525

““undo the base”undo the base” te 15.0ln)15100(

1525ln

t15.0)15100(

1525ln

t

)15100(

1525ln

15.0

1 min2.14t

Barking dog sound intensity: watts/sq meter Barking dog sound intensity: watts/sq meter

Sound Intensity

0

log10)(I

IIL

Loudness of the soundLoudness of the sound (in decibels) as a function(in decibels) as a function of the sound intensityof the sound intensity

Intensity of Intensity of the sound inthe sound in watts/sq meterwatts/sq meter

Intensity of soundIntensity of sound at the thresholdat the threshold of hearing ( of hearing ( watts per sq meter)watts per sq meter)

1210

410

How Loud is a dog’s bark?How Loud is a dog’s bark?

12

4

10

10log10)(

IL )12(410log10 810log10 db808*10

HOMEWORK

• Section 7-5

Page 510: Page 510: (evens) 4-42, 46-52, 76, 78(evens) 4-42, 46-52, 76, 78

(26 points) (26 points)