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Solving Exponential Logarithmic Equations

Exponentials

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Solving Exponential Logarithmic Equations

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    1. Solving Exponential & Logarithmic Equations Strategies and Practice

    2. Exponentials & Equal Bases 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 You try… 1. 4x = 83 2. 5x-2 = 25x 3. 6(3x+1) = 54

    3. Exponentials of Unequal Bases Use logarithm (inverse function) of same base on both sides of equation EX: Solve: ex = 72 ? lnex = ln72? xlne = ln72 x = ln72 (calc ready form) x ~ 4.277 EX: Solve: 7x-1 = 12? log77x-1 = log712 (x-1)log77 = log712 ? x-1 = log712 x = 1+log712 x ~ 1.277 You try… 4. Solve e2x = 5 5. Solve 32t-5 = 15

    4. Single Side Log Equations Convert to exponential (inverse) form EX: Solve: lnx = -1/2 ? e-1/2 = x ? .607 ~ x EX: Solve: 2log53x = 4 ? log53x = 2 52 = 3x ? 25/3 = x Use Laws to condense (domain solutions only) EX: Solve: log4x – log4(x-1) = ½ ? log4(x2-x)= ½ 41/2 = x2 – x ? 0 = x2-x-2 ?(x-2)(x+1)? x=2 You try… 6. lnx = -7 7. logx625 = 4 8. lnx+ln(x-3) = 1

    5. Double-Sided Log Equations Equate powers (domain solutions only) EX: Solve: log5(5x-1) = log5(x+7) 5x – 1 = x + 7 ? x = 2 EX: Solve: ln(x-2) + ln(2x-3) = 2lnx Use a property: ln(x-2)(2x-3) = lnx2 2x2 – 7x + 6 = x2 ? x2-7x+6=0 ? x = 6 & 1 You try… 9. Solve ln3x2 = lnx 10. Solve log2x+log2(x+5) = log2(x+4)

    6. 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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