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8-5 Exponential & 8-5 Exponential & Logarithmic EquationsLogarithmic Equations
Strategies and PracticeStrategies and Practice
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ObjectivesObjectives– Use like bases to solve exponential
equations.
– Use logarithms to solve exponential equations.
– Use the definition of a logarithm to solve
logarithmic equations.
– Use the one-to-one property of logarithms to solve logarithmic equations.
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Use like bases to solve Use like bases to solve exponential equationsexponential equations
• Equal bases must have equal exponents
EX: Given 3x-1 = 32x + 1
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If possible, rewrite to make bases equal
EX: Given 2-x = 4x+1
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Isolate function if needed—
3(2x)=48
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You try…
1. 4x = 83
2. 5x-2 = 25x
3. 6(3x+1) = 54
4. e–x2 = e-3x - 4
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Solving Logarithmic EquationsSolving Logarithmic Equations• Convert to exponential (inverse) form
EX: Solve: 2log53x = 4
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Now you try….Now you try….
• log x = 6
• log 5x = 3
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Solving Logarithmic EquationsSolving Logarithmic EquationsUse Properties of Logs to condense
EX: Solve: log4x + log4(x-1) = ½
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You try…You try…
Solve lnx+ln(x-3) = 1
Solve log x + log (x + 2) = log (x + 6)
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Double-Sided Log EquationsDouble-Sided Log Equations• Equate powers (domain solutions only)
EX: Solve: log5(5x-1) = log5(x+7)
EX: Solve: ln(x-2) + ln(2x-3) = 2lnx
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You try…You try…
1. Solve ln3x2 = lnx 2. Solve log6(3x + 14) – log6 5 = log6 2x
3. Solve log2x+log2(x+5) = log2(x+4)
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Exponentials of Unequal BasesExponentials of Unequal Bases• Use logarithm (inverse function) of same
base on both sides of equation
EX: Solve: ex = 72
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Now you try….Now you try….
• 2ex + 2 = 12
• ex – 9 = 19
• 7 - 2ex = 5
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• EX: Solve: 7x-1 = 12
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You try…You try…
1. Solve 3(2x) = 42
2. Solve 32t-5 = 15
3. Solve e2x = 5
4. Solve ex + 5 = 60
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Solving Logarithmic EquationsSolving Logarithmic Equations• Convert to exponential (inverse) form EX: Solve: lnx = -1/2
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Now you tryNow you try
• ln (2x – 1) = 0
• ln x = -3
• 3ln 5x = 10
Solve lnx+ln(x-3) = 1
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SUMMARYSUMMARY• Equal bases Equal exponents
• Unequal bases Apply log of given base
• Single side logs Convert to exp form
• Double-sided logs Equate powers
Note: Any solutions that result in a log(neg) cannot be used!