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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