solving exponential and logarithmic equations section 3.4 jmerrill, 2005 revised, 2008
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Solving Exponential and Solving Exponential and Logarithmic EquationsLogarithmic Equations
Section 3.4Section 3.4
JMerrill, 2005JMerrill, 2005
Revised, 2008Revised, 2008
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Same BaseSame Base
Solve: 4Solve: 4x-2 x-2 = 64= 64xx
44x-2x-2 = (4 = (433))xx
44x-2x-2 = 4 = 43x3x
x–2 = 3xx–2 = 3x -2 = 2x-2 = 2x -1 = x-1 = x
If bM = bN, then M = N64 = 43
If the bases are already =, just solve the exponents
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You DoYou Do
Solve 27Solve 27x+3x+3 = 9 = 9x-1x-1
x 3 x 13 2
3x 9 2x 2
3 3
3 3
3x 9 2x 2
x 9 2
x 11
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Review – Change Logs to Review – Change Logs to ExponentsExponents
loglog33x = 2x = 2
loglogxx16 = 216 = 2
log 1000 = xlog 1000 = x
32 = x, x = 9
x2 = 16, x = 4
10x = 1000, x = 3
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Using Properties to Solve Using Properties to Solve Logarithmic EquationsLogarithmic Equations
If the exponent is a variable, then take the If the exponent is a variable, then take the natural log of both sides of the equation and natural log of both sides of the equation and use the appropriate property. use the appropriate property.
Then solve for the variable.Then solve for the variable.
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Example: SolvingExample: Solving
22x x = 7= 7 problemproblem ln2ln2xx = ln7 = ln7 take ln both sidestake ln both sides xln2 = ln7xln2 = ln7 power rulepower rule x = x = divide to solve for divide to solve for
xx
x = 2.807x = 2.807
ln7ln2
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Example: SolvingExample: Solving
eex x = 72= 72 problemproblem lnelnexx = ln 72 = ln 72 take ln both sidestake ln both sides x lne = ln 72x lne = ln 72 power rulepower rule x = 4.277x = 4.277 solution: becausesolution: because
ln e = ?ln e = ?
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You Do: SolvingYou Do: Solving
2e2ex x + 8 = 20+ 8 = 20 problemproblem 2e2exx = 12 = 12 subtract 8subtract 8 eexx = 6 = 6 divide by 2divide by 2 ln eln exx = ln 6 = ln 6 take ln both sidestake ln both sides x lne = 1.792x lne = 1.792 power rulepower rule
x = 1.792x = 1.792 (remember: lne = (remember: lne = 1)1)
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ExampleExample
Solve 5Solve 5x-2x-2 = 4 = 42x+32x+3
ln5ln5x-2x-2 = ln4 = ln42x+32x+3
(x-2)ln5 = (2x+3)ln4(x-2)ln5 = (2x+3)ln4 The book wants you to distribute…The book wants you to distribute… Instead, divide by ln4Instead, divide by ln4 (x-2)1.1609 = 2x+3(x-2)1.1609 = 2x+3 1.1609x-2.3219 = 2x+31.1609x-2.3219 = 2x+3 x≈6.3424x≈6.3424
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Solving by Rewriting as an Solving by Rewriting as an ExponentialExponential
Solve logSolve log44(x+3) = 2(x+3) = 2
4422 = x+3 = x+3 16 = x+316 = x+3 13 = x13 = x
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You DoYou Do
Solve 3ln(2x) = 12Solve 3ln(2x) = 12 ln(2x) = 4ln(2x) = 4 Realize that our base is e, soRealize that our base is e, so ee44 = 2x = 2x x x ≈ 27.299≈ 27.299
You always need to check your answers You always need to check your answers because sometimes they don’t work!because sometimes they don’t work!
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Using Properties to Solve Using Properties to Solve Logarithmic EquationsLogarithmic Equations
1.1. Condense both sides first (if Condense both sides first (if necessary).necessary).
2.2. If the bases are the same on both If the bases are the same on both sides, you can cancel the logs on both sides, you can cancel the logs on both sides.sides.
3.3. Solve the simple equationSolve the simple equation
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Example: Solve for xExample: Solve for x
loglog336 = log6 = log333 + log3 + log33xx problemproblem
loglog336 = log6 = log333x3x condensecondense
6 = 3x6 = 3x drop logsdrop logs
2 = x2 = x solutionsolution
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You Do: Solve for xYou Do: Solve for x
log 16 = x log 2 log 16 = x log 2 problemproblem
log 16 = log 2log 16 = log 2xx condensecondense
16 = 2x16 = 2x drop logsdrop logs
x = 4 x = 4 solutionsolution
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You Do: Solve for xYou Do: Solve for x
loglog44x = logx = log4444 problemproblem
= log= log4444 condensecondense
= 4= 4 drop logsdrop logs
cube each sidecube each side
X = 64X = 64 solutionsolution
13
13
4log x
13x
3133 4x
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ExampleExample
7xlog7xlog225 = 3xlog5 = 3xlog225 + ½ log5 + ½ log222525
loglog22557x7x = log = log22553x3x + log + log2225 25 ½ ½
loglog22557x7x = log = log22553x3x + log + log225511
7x = 3x + 17x = 3x + 1 4x = 14x = 1
14
x
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You DoYou Do
Solve:Solve: loglog777 + log7 + log772 = log2 = log77x + logx + log77(5x – 3) (5x – 3)
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You Do AnswerYou Do Answer
Solve:Solve: loglog777 + log7 + log772 = log2 = log77x + logx + log77(5x – 3)(5x – 3)
loglog7714 = log14 = log7 7 x(5x – 3)x(5x – 3)
14 = 5x14 = 5x22 -3x -3x 0 = 5x0 = 5x22 – 3x – 14 – 3x – 14 0 = (5x + 7)(x – 2)0 = (5x + 7)(x – 2) 7
,25
x
Do both answers work? NO!!
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Final ExampleFinal Example
How long will it take for $25,000 to grow to How long will it take for $25,000 to grow to $500,000 at 9% annual interest $500,000 at 9% annual interest compounded monthly?compounded monthly?
0( ) 1
ntrA t A
n
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ExampleExample
0( ) 1
ntrA t A
n12
0.09500,000 25,000 1
12
t
1220 1.0075 t
12t ln(1.0075) ln20
ln20t
12ln1.0075t 33.4