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5-4 Exponential & Logarithmic Equations

5-4 Exponential & Logarithmic Equations. Strategies and Practice. Objectives. – Use like bases to solve exponential equations. – Use logarithms to solve exponential equations. – Use the definition of a logarithm to solve logarithmic equations.

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5-4 Exponential & Logarithmic Equations

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  1. 5-4 Exponential & Logarithmic Equations Strategies and Practice

  2. 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.

  3. Use like bases to solve exponential equations • Equal bases must have equal exponents EX: Given 3x-1 = 32x + 1thenx-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

  4. You try… 1. 4x = 83 2. 5x-2 = 25x 3. 6(3x+1) = 54 4. e–x2 = e-3x - 4

  5. 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

  6. You try… 1. Solve 3(2x) = 42 2. Solve 32t-5 = 15 3. Solve e2x = 5 4. Solve ex + 5= 60

  7. Solving Logarithmic Equations • Convert to exponential (inverse) form EX: Solve: lnx = -1/2 elnx = e-1/2  x = e-1/2  x  .607 EX: Solve: 2log53x = 4  log53x = 2 (get the log by itself) 52 = 3x  25/3 = x Use Properties of Logs to condense 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

  8. You try… 1. Solve lnx = -7 2. Solve 2log5 3x = 4 3. Solve . lnx+ln(x-3) = 1 4. Solve . 5 + 2ln x = 4

  9. 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

  10. You try… 1. Solve ln3x2 = lnx 2. Solve log6(3x + 14) – log6 5 = log6 2x 3. Solve log2x+log2(x+5) =log2(x+4)

  11. 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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