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8-5 Exponential & Logarithmic Equations

8-5 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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8-5 Exponential & Logarithmic Equations

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  1. 8-5 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 + 1

  4. If possible, rewrite to make bases equal EX: Given 2-x = 4x+1

  5. Isolate function if needed— 3(2x)=48

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

  7. Solving Logarithmic Equations • Convert to exponential (inverse) form EX: Solve: 2log53x = 4

  8. Now you try…. • log x = 6 • log 5x = 3

  9. Solving Logarithmic Equations Use Properties of Logs to condense EX: Solve: log4x + log4(x-1) = ½

  10. You try… Solve lnx+ln(x-3) = 1 Solve log x + log (x + 2) = log (x + 6)

  11. Double-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

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

  13. Exponentials of Unequal Bases • Use logarithm (inverse function) of same base on both sides of equation EX: Solve: ex = 72

  14. Now you try…. • 2ex+2 = 12 • ex – 9 = 19 • 7 - 2ex = 5

  15. EX: Solve: 7x-1 = 12

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

  17. Solving Logarithmic Equations • Convert to exponential (inverse) form EX: Solve: lnx = -1/2

  18. Now you try • ln (2x – 1) = 0 • ln x = -3 • 3ln 5x = 10 Solve lnx+ln(x-3) = 1

  19. 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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