CP Math CP Math 10-510-5

Solve Exponential and Logarithmic Solve Exponential and Logarithmic EquationsEquations

Quiz 7-5:Quiz 7-5:

1.1.

2.2.

4.4.

5.5.

3.3.

6.6.

ExpandExpand

18log4

5

4log

2log5log 44

CondenseCondense

xyx 55 log7log3

3

4

3logy

x 7log8

What is the decimal equivalent?What is the decimal equivalent?

What you’ll learnWhat you’ll learnReview of Inverse FunctionsReview of Inverse Functions

… and whyLogarithms are used extensively in science. Solving Logarithms are used extensively in science. Solving equations is how problems in science are solved.equations is how problems in science are solved.

Using Logarithms to solve exponential equationsUsing Logarithms to solve exponential equations

Property of EqualityProperty of Equality

Using exponents to solve logarithmic equationsUsing exponents to solve logarithmic equations

Real World Problems: Real World Problems: Newton’s Law of coolingNewton’s Law of coolingDistance a tornado stays on the groundDistance a tornado stays on the groundThe apparent “magnitude” of a starThe apparent “magnitude” of a star

2log2

12log

2log

Your turn:Your turn:

1.1.

Use the change of base formula to find:Use the change of base formula to find:

2.2. 4log7

7124.07log

4log

Solving Equations ReviewSolving Equations Review

5423 xx

Single Variable Equation:Single Variable Equation: ““Isolate the variable”Isolate the variable”

subtract subtract 3x3x from both sides from both sides

52 x-3x-3x -3x-3x

add add 55 to both sides to both sides

+5+5 +5+5

7x

What does “solve” mean?

Solving Equations ReviewSolving Equations ReviewRadical Equation:Radical Equation:

3123 x

““Isolate the radical”Isolate the radical”

““Undo the radical”Undo the radical”

add add 11 to both sides to both sides

+1+1 +1+1

423 x square both sidessquare both sides

22423 x 1623 x add add 22 to both sides to both sides

+2+2 +2+2

183 x divide both divide both sides by 3sides by 3

6x

Solving an Exponential Equation: The Solving an Exponential Equation: The easiest problemeasiest problem

xx 422 ““Isolate the power, Undo the power”Isolate the power, Undo the power”

What is the inverse functionWhat is the inverse function of power base 2?of power base 2?

xx 422 2log2log

xx 4

2log42log 22 xx Logarithm exponent propertyLogarithm exponent property

12log

2log2log2 Remember:Remember:

42 x 2x

Another exampleAnother example

xx 41312 77 ““Isolate the power, Undo the power”Isolate the power, Undo the power”

Apply inverse function of “power Apply inverse function of “power base 7”base 7”

xx 41312

7log4137log)12( 77 xx Logarithm Logarithm exponent exponent propertyproperty

17log

7log7log7 Remember:Remember:

1316 x

126 x

xx 4137

127 7log7log

+4x +4x+4x +4x

-1 -1-1 -12x

Your turn:Your turn:SolveSolve::

4.4.

““Isolate the power” thenIsolate the power” then “ “undo the power”undo the power”

3.3.

1014 44 xx

xx 282 33

Changing the base of a Changing the base of a powerpower

31 28

21 24

EasyEasy

HarderHarder

222 )2(4 42222 )3(9 43xx 222 )3(9 x43

Your turn:Your turn:Change the base of the power as indicatedChange the base of the power as indicated::

6.6.

5.5.

?2 416

?1 327

7.7.?2 525 x

Solving using “convert to same Solving using “convert to same base”base”

““Undo the exponent”Undo the exponent”12 279 xx

If I take log base 9 of each side it won’t eliminate BOTH basesIf I take log base 9 of each side it won’t eliminate BOTH bases

1322 33

xx

)1(32*2 33 xx

Can the two bases be rewritten as Can the two bases be rewritten as a power of the same base?a power of the same base?

Power of a power rulePower of a power rule

334 33 xx Distributive PropertyDistributive Property

333

43 3log3log xx Property of equalityProperty of equality

334 xx 3x

Solving using “power” of the power Solving using “power” of the power rulerule

Take natural log of both sideTake natural log of both side12 279 xx

12 27ln9ln xxpower rulepower rule

simplifysimplify

3x27ln)1(9ln2 xx

9ln

27ln)1(2 xx

÷ ln 9 ÷ ln 9÷ ln 9 ÷ ln 9

)5.1)(1(2 xx

5.15.12 xx

simplifysimplify

-1.5x -1.5x-1.5x -1.5x

5.15.0 xx by 2 x by 2x by 2 x by 2

22 48 xx

2223 22

xx

8 and 4 are 8 and 4 are bothboth powers of 2. powers of 2.

Power of a Power propertyPower of a Power property

4263 22 xx ““undo the power”undo the power”

4263 xx-2x -2x-2x -2x

46 x-6 -6-6 -6

10x

Did I do a step mentally?Did I do a step mentally?

Solving using “convert to same Solving using “convert to same base”base”

22 48 xx Take natural log of both sidesTake natural log of both sides

Power propertyPower property

-0.667x -0.667x-0.667x -0.667x

10x

-2 -2-2 -2

Solving using the “power” of the Solving using the “power” of the power rulepower rule

22 4ln8ln xx

4ln)2(8ln)2( xx÷ ln 8 ln 8÷ ln 8 ln 8

8ln

4ln)2(2 xx

667.0)2(2 xx

333.1667.02 xx

333.12333.0 x

333.3333.0 x÷ 0.333 ÷0.333÷ 0.333 ÷0.333

Your turn:Your turn:SolveSolve::

9.9.

13525 xx

““Isolate the power” thenIsolate the power” then “ “undo the power”undo the power”

8.8.

xx 93 12

10.10. 52 215 xx

Solve using “undo the power”Solve using “undo the power”

““Isolate the power”Isolate the power”

““Undo the power”Undo the power”

753 12 x

-5 -5-5 -5

2log12 3x Change of baseChange of base formulaformula

23 12 x

2log3log 312

3 x

3ln

2ln12 x

+1 +1+1 +1

÷2 ÷2÷2 ÷2

815.0x

63093.012 x

63093.12 x

Solve using the “power” of the Solve using the “power” of the power rulepower rule

““Isolate the power”Isolate the power”

Natural log left/rightNatural log left/right

753 12 x

-5 -5-5 -5

2ln3ln)12( x

23 12 x

2ln3ln 12 x

3ln

2ln12 x

+1 +1+1 +1

÷2 ÷2÷2 ÷2

815.0x

63093.012 x

63093.12 x

Your turn:Your turn:Solve:Solve:

11.11.

12.12.

16583 x

1025 4 x

Natural Logarithm FunctionNatural Logarithm Function

xxf ln)(

What is the domain?What is the domain?

Does NOT make sense.Does NOT make sense.

?)5ln(

0x

When solving logarithm When solving logarithm equationsequations

Some solutions won’t make sense.Some solutions won’t make sense.

Extraneous solutionExtraneous solution: an apparent solution that does not : an apparent solution that does not work when plugged back into the original equation.work when plugged back into the original equation.

You You MUSTMUST check the solutions in the original equation. check the solutions in the original equation.

Solving Logarithmic Solving Logarithmic EquationsEquations

““Isolate the logarithm”Isolate the logarithm”7loglog 22 x““undo the logarithm”undo the logarithm”

7loglog 22 22 x Inverse of log base 2 Inverse of log base 2 is exponent base 2.is exponent base 2.

x = 7

7log7log 22 Plug back in to check!Plug back in to check!

Why do we need to check? Why do we need to check? Checks!Checks!

Remember this:Remember this:

For a log equation: if the solution results in the log ofFor a log equation: if the solution results in the log of a negative number, that number is NOT a solution.a negative number, that number is NOT a solution.

sense no makes )10ln(

Solving Logarithmic Solving Logarithmic EquationsEquations

““Isolate the logarithm”Isolate the logarithm”)5(log)74(log 55 xx““undo the logarithm”undo the logarithm”

)5(log)74(log 55 55 xxInverse of log base 5 Inverse of log base 5 is exponent base 5.is exponent base 5.

4x - 7 = x + 5

3x = 12

x = 4

Subtract ‘x’ from both sides.Subtract ‘x’ from both sides.

Divide both sides by 3Divide both sides by 3

Plug back in to check!Plug back in to check!)54(log)74*4(log 55

9log9log 55 ChecksChecks

Your Turn:Your Turn:

13.13.

14.14.

)11ln()74ln( xx

Solve:Solve:

)93(log)72(log 33 xx

Remember to check you solutions by plugging the solutionRemember to check you solutions by plugging the solution for ‘x’ back into the original equation.for ‘x’ back into the original equation.

Good to here with 5Good to here with 5thth period. period.

Solving Logarithmic Solving Logarithmic EquationsEquations

55log2 x

52ln

5lnx

Power property of logarithmsPower property of logarithms

Change of base Change of base

55log2 x

5ln

2ln5x Use inverse property of multiplicationUse inverse property of multiplication

)4307.0(5x 1534.2x

Your turn:Your turn:Solve:Solve:

“ “isolate the log” then isolate the log” then “ “undo the log”undo the log”

15.15. 45log2 x

16.16. 64log 53 x Don’t forget the power Don’t forget the power

property for logarithms.property for logarithms.

More complicated Logarithmic More complicated Logarithmic EquationsEquations

““Isolate the logarithm”Isolate the logarithm”75log2 22 x

““undo the logarithm”undo the logarithm”

52ln

5ln)2( x

Power property of logarithmsPower property of logarithms

Add ‘2’ to both sides.Add ‘2’ to both sides.

Change of base Change of base

55log 22 x

-2 -2-2 -2

55log)2( 2 x

5ln

2ln52 x

Use inverse property of multiplicationUse inverse property of multiplication

5ln

2ln52 x

)4307.0(52 x 1524.4x

Your turn:Your turn:Solve:Solve:

“ “isolate the log” then isolate the log” then “ “undo the log”undo the log”

17.17. 63log3 124 x

18.18. 54ln27 3 x Don’t forget the power Don’t forget the power property for logarithms.property for logarithms.

Solving Logarithmic Solving Logarithmic EquationsEquations ““Isolate the logarithm”Isolate the logarithm”

3)15(log4 x ““undo the logarithm”undo the logarithm”

3)15(log 44 4 x Inverse of log base 4 Inverse of log base 4 is exponent base 4.is exponent base 4.

5x - 1 = 64

655 xSubtract ‘1’ from both sidesSubtract ‘1’ from both sides

Divide both sides by ‘5’Divide both sides by ‘5’

x = 13 Plug back in to check!Plug back in to check!

ChecksChecks

3)113*5(log4

364log4

Your Turn:Your Turn:19.19.

Solve:Solve:

35ln5 x

6)32(log2 3 x20.20.

Solving Logs requiring condensing the Solving Logs requiring condensing the product. product.

““Isolate the logarithm”Isolate the logarithm”2)5log(2log xx““undo the logarithm”undo the logarithm”

2)5(2log 1010 10 xx

““condense the product”condense the product”

2x(x - 5) = 100

Inverse of log base 10 Inverse of log base 10 is exponent base 10is exponent base 10

Quadratic Quadratic put in standard form put in standard form

Divide both sides by ‘2’Divide both sides by ‘2’

factorfactor

2)5(2log xx

0100102 2 xx05052 xx0)5)(10( xx

x = 10, -5 Zero factor propertyZero factor property

Check the solution:Check the solution:2)5log(2log xx x = 10, -5

2)510log()10*2log(

2)5log()20log(

2)5*20log(

““Condense the product”Condense the product”

2100log

100102

Convert to exponent formConvert to exponent form

ChecksChecks

Check the solution:Check the solution:2)5log(2log xx x = 10, -5

2)55log())5(2log(

2)10log()10log(

Or, convert to exponent formOr, convert to exponent form

This doesn’t make sense This doesn’t make sense punch punch this into your calculator.this into your calculator.

)10log(

1010 x

There is NO exponent that will cause a There is NO exponent that will cause a positive number to equal a negative number.positive number to equal a negative number.

-5 is NOT-5 is NOT a solutiona solution

Eliminate extraneous solutions Eliminate extraneous solutions using the DOMAIN:using the DOMAIN:

2)5log(2log xx x = 10, -5

What is the domain?What is the domain?

-5 is NOT-5 is NOT a solutiona solution

0x 5x

You cannot take the log of a negative number!You cannot take the log of a negative number!

You cannot take the log of zero!You cannot take the log of zero!

You can only take the log of You can only take the log of positive numberpositive number!!

2)5log(2log xx

Which condition is more restrictive?Which condition is more restrictive?

Your Turn:Your Turn:

21.21.

22.22.

25ln5ln 22 x

63log4log 22 x

““Undo the exponent”Undo the exponent”

Solving Logarithmic equations by Solving Logarithmic equations by graphinggraphing

2)5log(2log xx x = 10

Set equal to zero (-2 each side)Set equal to zero (-2 each side))5log(2log20 xx

Replace ‘0’ with ‘y’Replace ‘0’ with ‘y’)5log(2log2 xxySolution: the x-value that causes ‘y’ to equal ‘0’, Solution: the x-value that causes ‘y’ to equal ‘0’, x-interceptsx-intercepts

graphgraph Adjust windowAdjust window

Newton’s Law of CoolingNewton’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.

rtsos eTTTtT )()(

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

Surrounding Surrounding TemperatureTemperature

Initial TempInitial Temp of the object of the object

(temperature at t = 0)(temperature at t = 0)

Cooling rateCooling rate

TimeTime

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. r = 0.15 When will the egg be 25C. r = 0.15 When will the egg be 25ººC?C?

rtsos eTTTtT )()( te 15.0)15100(1525

““isolate the power”isolate the power”

te 15.08510

““undo the power”undo the power”

te 15.0ln1176.0ln

t15.014.2 min2.14t

Subtract 15Subtract 15 from both sidesfrom both sides

te 15.01176.0 Divide both Divide both sides by 85sides by 85

““undo power” undo power” Natural log both Natural log both sidessides

Divide both sides by -0.15Divide both sides by -0.15

Your Turn:Your Turn:You are baking a cake. When you take the cake out of the oven, You are baking a cake. When you take the cake out of the oven, it is at 350ºF. The room temperature is 70it is at 350ºF. The room temperature is 70ºF. The cooling ºF. The cooling rate is rate is 0.08. How many minutes will it take for the cake cool to 1000.08. How many minutes will it take for the cake cool to 100ºF?ºF?

rtsos eTTTtT )()(

23. 23. Solve for ‘time’Solve for ‘time’

te 08.0)70350(70100 Plug #’s into the formula.Plug #’s into the formula.

Isolate the powerIsolate the power

te 08.0

)280(

30

te 08.0ln107.0ln

Undo the powerUndo the power

t08.02336.2 Solve for ‘t’Solve for ‘t’

t

08.0

2336.2 t = 27.9 minutest = 27.9 minutes

Using this idea to solve Using this idea to solve equationsequations

313 636 xx

3132 66 xx

Replace 36 with a power with base ‘6’Replace 36 with a power with base ‘6’::

Power of a Power propertyPower of a Power property

326 66 xx ““undo the power”undo the power”

36

266 6log6log xx

326 xx-x -x-x -x

325 x-2 -2-2 -2

15 x

51x

Solve using “undo the power”Solve using “undo the power”

““Isolate the power”Isolate the power”

““Undo the power”Undo the power”

528 x+2+2 +2+2

78 x

7log8log 88 x

7log8x8log

7logxChange of baseChange of base

formulaformula

9358.0x

Solve using the “power” of the Solve using the “power” of the power rulepower rule

““Isolate the power”Isolate the power”

Natural log left/rightNatural log left/right

528 x+2+2 +2+2

78 x

7ln8ln x

7ln8ln x 8ln

7lnx 9358.0x

Power rulePower rule