Quiz 3-4Quiz 3-4

?)16log( 5 xy

1.1. Convert to a common Convert to a common logarithmlogarithm

x27log4

2. Expand:2. Expand:

zyx ln)lnln3( 3. Condense:3. Condense:

272 x

yx log5log4log2

z

yx3

ln

3.53.5

Equation Solving and ModelingEquation Solving and Modeling

What you’ll learn aboutWhat you’ll learn about Solving Exponential EquationsSolving Exponential Equations Solving Logarithmic EquationsSolving Logarithmic Equations Orders of Magnitude and Logarithmic ModelsOrders of Magnitude and Logarithmic Models Newton’s Law of CoolingNewton’s Law of Cooling Logarithmic Re-expressionLogarithmic Re-expression

… … and whyand whyThe Richter scale, pH, and Newton’s Law of Cooling, The Richter scale, pH, and Newton’s Law of Cooling,

are examples of real world phenomena that are are examples of real world phenomena that are modeled by logarithmic and exponential functions.modeled by logarithmic and exponential functions.

One-to-One PropertiesOne-to-One Properties

The book teaches this “new” property. I hate new propertiesThe book teaches this “new” property. I hate new properties to remember if I don’t need them!!!!to remember if I don’t need them!!!!

vu bb ““undo” base ‘b’undo” base ‘b’v

bu

b bb loglog vu Why memorize a new property if you can get the same resultWhy memorize a new property if you can get the same result by just using properties already learned?by just using properties already learned?

One-to-One propertyOne-to-One property: for any exponential function: : for any exponential function: xbxf )(

If then: u = v If then: u = v vu bb

One-to-One PropertiesOne-to-One Properties

The book teaches this “new” property. I hate new propertiesThe book teaches this “new” property. I hate new properties to remember if I don’t need them!!!!to remember if I don’t need them!!!!

““undo” log base ‘b’undo” log base ‘b’xu bb bb loglog vu

Why memorize a new property if you can get the same resultWhy memorize a new property if you can get the same result by just using properties already learned?by just using properties already learned?

xxf blog)(

If then: u = v If then: u = v vu bb loglog One-to-One propertyOne-to-One property: for any logarithm function: : for any logarithm function:

vu bb loglog

Solving an Exponential Solving an Exponential Equation AlgebraicallyEquation Algebraically

52

140

2

x

Isolate the base and the exponent:Isolate the base and the exponent:

32

2

1

2

1

x

32

x

6x

““undo” base ½ undo” base ½

Solve:Solve:

8

1

2

1 2

x

Can you rewrite 1/8 as a power of ½?Can you rewrite 1/8 as a power of ½?

3

21

2

21 2

1log

2

1log

x

Your Turn: Your Turn: solve for xsolve for x

1. 1. 43

130

5

x

2. 2. 20102 3

x

Solving a Logarithmic Solving a Logarithmic EquationEquation

3log 3 x

33 10x

3log 10103

x

10x

““undo” log base 10undo” log base 10

Solve for ‘x’Solve for ‘x’

Your turn:Your turn:

3.3. 2log 2 x

4.4. x216log3

Solving a Logarithmic Solving a Logarithmic EquationEquation

54)3ln(2 x

1)3ln(2 x

Isolate the varable: Subtract 4 Isolate the varable: Subtract 4 from both sidesfrom both sides

Divide both sides by 2Divide both sides by 2

21)3ln( x ““undo” log base ‘e’undo” log base ‘e’

21)3ln( ee x 2

13 ex

21

3 ex

Solve for ‘x’Solve for ‘x’

Your Turn:Your Turn:

5.5. 5)2log(3 x

6.6. 6.198.0 x

Orders of MagnitudeOrders of MagnitudeThe common logarithm of a positive quantity is its The common logarithm of a positive quantity is its order of order of

magnitudemagnitude..

DistanceDistance: Mercury to the sun: Mercury to the sun

DistanceDistance: Pluto to the sun: Pluto to the sun

meters91079.5 meters12109.5

What is the What is the order of magnitude order of magnitude differencedifference when comparing when comparing the distance of Mercury and Pluto from the sun?the distance of Mercury and Pluto from the sun?

77.1210log9.5log)109.5log( 1212

763.1010log79.5log)1079.5log( 1010

007.2763.1077.12 About 2 orders of Mag.About 2 orders of Mag.

Orders of MagnitudeOrders of MagnitudeThe common logarithm of a positive quantity is its The common logarithm of a positive quantity is its order of order of

magnitudemagnitude..

What is the What is the order of magnitude order of magnitude differencedifference when comparing when comparing the distance of Mercury and Pluto from the sun?the distance of Mercury and Pluto from the sun?

77.12)109.5log( 12 763.10)1079.5log( 10

007.2763.1077.12

Order of magnitudeOrder of magnitude: common log of the number: common log of the number

Difference in order of magnitude between two numbersDifference in order of magnitude between two numbers:: difference (subtract) the common logs of the two numbersdifference (subtract) the common logs of the two numbers

Your Turn:Your Turn:

7. 7. What is the order of magnitude difference What is the order of magnitude difference between a gram and a kilogram?between a gram and a kilogram?

8. 8. What is the order of magnitude difference between What is the order of magnitude difference between a person’s $100,000 salary and the $1 Trillion thea person’s $100,000 salary and the $1 Trillion the government spent that it didn’t have last year? government spent that it didn’t have last year?

Earthquake IntensityEarthquake IntensityIs measured by the amplitude of the vibrationIs measured by the amplitude of the vibration felt at the measuring station.felt at the measuring station.

BT

amplitudeR log

T: T: PeriodPeriod of the seismic wave (seconds) of the seismic wave (seconds)

B: a “fudge factor” to account for weakening of the B: a “fudge factor” to account for weakening of the seismic wave from origin to where it is measured.seismic wave from origin to where it is measured.

Amplitude is measured in Amplitude is measured in metersm 610

Comparing Orders of Comparing Orders of Magnitude of Earthquake Magnitude of Earthquake IntensitiesIntensities

BT

amplitudeR log

2010 Gugerat, India: R = 7.92010 Gugerat, India: R = 7.9

1999 Athens, Greece1999 Athens, Greece R = 5.9R = 5.9

We want to compare the We want to compare the amplitudesamplitudes..

BT

agugerat log9.7 BT

aathens log9.5

B

T

aB

T

aathensgugurat loglog9.59.7

Comparing Orders of Comparing Orders of Magnitude of Earthquake Magnitude of Earthquake IntensitiesIntensities

B

T

aB

T

aathensgugurat loglog9.59.7

athens

gugurat

a

alog2

athens

gugurat

a

a210

100 times greater !!100 times greater !!

7.9 – 5.9 is the difference in magnitudes (difference 7.9 – 5.9 is the difference in magnitudes (difference between the natural logs of two different numbers).between the natural logs of two different numbers).

The difference in magnitudes is the power of 10 differenceThe difference in magnitudes is the power of 10 difference between the two different numbers).between the two different numbers).

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

pH = -log [HpH = -log [H++]]

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

pH is the magitude (common log) of the hydrogen ion.pH is the magitude (common log) of the hydrogen ion.

The difference in pH between two solutions is the is the power The difference in pH between two solutions is the is the power of 10 difference between their hydrogen ion concentrations. of 10 difference between their hydrogen ion concentrations.

Your Turn:Your Turn:9. 9. The pH of carbonated water is 3.9 and the pH of The pH of carbonated water is 3.9 and the pH of household ammonia is 11.9. What are their hydrogenhousehold ammonia is 11.9. What are their hydrogen ion concentrations? ion concentrations?

10. 10. For problem #9, how many times greater is the For problem #9, how many times greater is the hydrogen ion concetration of carbonate water thathydrogen ion concetration of carbonate water that that of ammonia? that of ammonia?

pH = -log [HpH = -log [H++]]

11. 11. By how many orders of magnitude do the concentrationsBy how many orders of magnitude do the concentrations differ? differ?

Newton’s Law of CoolingNewton’s Law of Cooling

ktmediumoriginalmedium eTTTtT )()(

ktmom eTTTtT )()(

is the temperature of the surrounding mediumis the temperature of the surrounding mediummT

oT is the original temperature of the objectis the original temperature of the object

Isaac Newton discovered that a heated object will cool at a rateIsaac Newton discovered that a heated object will cool at a rate that is dependent upon the tempurature of the surrounding that is dependent upon the tempurature of the surrounding medium. The object’s temperature can be modeled by: medium. The object’s temperature can be modeled by:

t is timeis time

k is a “fudge factor” for cooling time is a “fudge factor” for cooling time

Example Newton’s Law of Example Newton’s Law of CoolingCooling

A hard-boiled egg at temperature 100A hard-boiled egg at temperature 100ººC is placed in 15C is placed in 15ººC C water to cool. Five minutes later the temperature of the water to cool. Five minutes later the temperature of the

egg is 55egg is 55ººC. When will the egg be 25C. When will the egg be 25ººC?C? kt

mom eTTTtT )()( T(5) = 55T(5) = 5515mT 100oT

We must first calculate “k” the “fudge factor” for this specificWe must first calculate “k” the “fudge factor” for this specific situation.situation.

5)15100(1555 ke

5

)15100(

1555 ke

Isolate the variableIsolate the variable

54706.0 ke

k5)4706.0ln( 1508.0k

Example Newton’s Law of Example Newton’s Law of CoolingCooling

A hard-boiled egg at temperature 100A hard-boiled egg at temperature 100ººC is placed in 15C is placed in 15ººC C water to cool. Five minutes later the temperature of the water to cool. Five minutes later the temperature of the

egg is 55egg is 55ººC. When will the egg be 25C. When will the egg be 25ººC?C? kt

mom eTTTtT )()( T(5) = 55T(5) = 5515mT 100oT

Now we know the complete modeling equation we can findNow we know the complete modeling equation we can find the time to reach 25 degrees the time to reach 25 degrees T(t) = 25 T(t) = 25

te )1508.0()15100(1525

te )1508.0(

)15100(

1525

Isolate the variableIsolate the variable

te 1508.01176.0

t1508.0)1176.0ln(

min 2.14t

Your turn:Your turn:A cake taken out of the oven at temperature of 350A cake taken out of the oven at temperature of 350 ºº F. It is placed on in F. It is placed on in a room with an ambient temperature of 70a room with an ambient temperature of 70ººF to cool. Ten minutes later F to cool. Ten minutes later the temperature of the cake is 150the temperature of the cake is 150ººF. When will the cake be cool F. When will the cake be cool

enough to put the frosting on (90enough to put the frosting on (90ººF) ?F) ?

ktmom eTTTtT )()( ?mT ?oT

12. 12. Calculate ‘k’ using the time to cool to the Calculate ‘k’ using the time to cool to the intermediate temperature.intermediate temperature.

13. 13. Once you have ‘k’, find ‘t’ for T(t) = desired temperature.Once you have ‘k’, find ‘t’ for T(t) = desired temperature.

Exponential Equations:Exponential Equations:

This looks like a quadratic equation!This looks like a quadratic equation!

0232 xx ee

Let: Let: xex 0232 xx 0)1)(2( xx

2x 1x2xe 1xe

2lnx 1lnx693.0x 0x

Your turn:Your turn:4

2

xx ee14. 14. Solve for ‘x’.Solve for ‘x’.

Hint: multiply left/right by then rewrite as a “quadratic”Hint: multiply left/right by then rewrite as a “quadratic” then solve. then solve.

xe

HOMEWORKHOMEWORK

Section 3-5: even problems:Section 3-5: even problems:2-8, 12-18, 26-36, 40-50.2-8, 12-18, 26-36, 40-50.(20 problems)(20 problems)

Don’t worry about the Don’t worry about the following slidesfollowing slides

Regression Models Related Regression Models Related by Logarithmic Re-by Logarithmic Re-ExpressionExpression

Linear regression:Linear regression: y = ax + by = ax + b

Natural logarithmic regression:Natural logarithmic regression: y = a + b*y = a + b*lnlnxx

Exponential regression:Exponential regression: y = ay = a·b·bxx

Power regression:Power regression: y = y = aa·x·xbb

Selecting Regression ModelSelecting Regression Modelx 1 2 3 4 5 6x 1 2 3 4 5 6y 2 5 10 17 26 38 y 2 5 10 17 26 38

1. Plot the data.1. Plot the data.

2.2. Based upon the plot, it looks like an Based upon the plot, it looks like an exponential or power functionexponential or power function

3.3. Use STAT regression feature to find the Use STAT regression feature to find the equation using each type of function.equation using each type of function.

4.4. Graph the curve Graph the curve andand the data points to see the data points to see how good the fit is. how good the fit is.

Three Types of Logarithmic Three Types of Logarithmic Re-ExpressionRe-Expression

Three Types of Logarithmic Three Types of Logarithmic Re-Expression (cont’d)Re-Expression (cont’d)

Three Types of Three Types of Logarithmic Re-ExpressionLogarithmic Re-Expression(cont’d)(cont’d)