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**5-4 Exponential & Logarithmic Equations**

Strategies and Practice

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Objectives – 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 exponential equations**

Equal bases must have equal exponents EX: Given 3x-1 = 32x + 1 then x-1 = 2x+1 x = -2 If possible, rewrite to make bases equal EX: Given 2-x = 4x+1 rewrite 4 as 22 2-x = 22x+2 then –x=2x+2 x=-2/3 Note: Isolate function if needed 3(2x)=48 2x =16

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You try… 1. 4x = 83 (22)x = (23)3 (2)2x = (2)9 So 2x = 9, or x = 4.5 2. 5x-2 = 25x 5x-2 = (52)x 5x-2 = (5)2x So x - 2 = 2x, or x = -2 3. 6(3x+1) = 54 3x+1 = 9 3x+1 = 32 So x + 1= 2, or x = 1 4. e–x2 = e-3x - 4 So –x2 = -3x & x2 – 3x – 4 = 0 & (x-4)(x+1) = 0 & x=4 and x = -1

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**Exponentials of Unequal Bases**

Use logarithm (inverse function) of same base on both sides of equation Solve: ex = 72 loge ex = loge x = ln 72 4.277 Solve: x-1 = 12 log7 7x-1 = log7 x - 1 = log7 12 x = log log 12 log 7 2.277 x =

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**You try… 1. Solve 3(2x) = 42 x = log2 14 3.807 2. Solve 32t-5 = 15**

t = 1/2(log ) 3.732 3. Solve e2x = 5 x = 1/2 ln 5 0.805 4. Solve ex + 5 = 60 x = ln 55 4.007

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**Solving Logarithmic Equations**

Rewrite into exponential form EX: Solve: ln x = - 1/2 loge x = - 1/2 e -1/2 = x 0.607 x = e -1/2 EX: Solve: log5 3x = 4 log5 3x = 2 52= 3x 25= 3x 25/3= x x = 25/3 8.333

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**Solving Logarithmic Equations**

Use properties of logarithms to condense. EX: Solve: log4x + log4(x-1) = ½ log4 x(x – 1) = 1/2 Check for extraneous roots. 4 1/2 = x(x – 1) 2 = x2 – x 0 = x2 – x – 2 0 = (x – 2)(x + 1) x = 2 & x = -1

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**You try… 1. Solve ln x = -7 x = e-7 0.000912 2. Solve 2 log3 2x = 4**

3. Solve ln x + ln (x-3) = 0 3 + 13 2 3 - 13 2 x = & x = 4. Solve ln x = 4 x = e-1/2 0.607

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**Double-Sided Log Equations**

Equate powers (domain solutions only) EX: Solve: log5(5x-1) = log5(x+7) 5x – 1 = x + 7 4x = 8 x = 2 EX: Solve: ln(x-2) + ln(2x-3) = 2lnx ln (x-2)(2x-3) = ln x2 Use the properties to condense. Check for extraneous roots. (x-2)(2x-3) = x2 2x2 – 7x + 6 = x2 x2 – 7x + 6 = 0 (x – 6)(x – 1)= 0 x = 6 & x = 1

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**You try… 1. Solve ln3x2 = lnx x = 0 & x = 1/3**

2. Solve log6(3x + 14) – log6 5 = log6 2x x = 2 3. Solve log2x+log2(x+5) = log2(x+4) x = 2 & x = 2

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**SUMMARY 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!

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