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Warm-Up

4 minutes. Warm-Up. Solve each equation for x. Round your answers to the nearest hundredth. 1) 10 x = 1.498. 2) 10 x = 0.0054. Find the value of x in each equation. 3) x = log 4 1. 4) ½ = log 9 x. 6.4.1 Properties of Logarithmic Functions. Objectives:

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Warm-Up

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  1. 4 minutes Warm-Up Solve each equation for x. Round your answers to the nearest hundredth. 1) 10x = 1.498 2) 10x = 0.0054 Find the value of x in each equation. 3) x = log4 1 4) ½ = log9 x

  2. 6.4.1 Properties of Logarithmic Functions Objectives: Simplify and evaluate expressions involving logarithms Solve equations involving logarithms

  3. Properties of Logarithms For m > 0, n > 0, b > 0, and b  1: Product Property logb (mn) = logb m + logb n

  4. Example 1 given: log5 12  1.5440 log5 10  1.4307 log5 120 = log5 (12)(10) = log5 12 + log5 10 1.5440 + 1.4307 2.9747

  5. logb = logb m – logb n m n Properties of Logarithms For m > 0, n > 0, b > 0, and b  1: Quotient Property

  6. 12 = log5 10 Example 2 given: log5 12  1.5440 log5 10  1.4307 log5 1.2 = log5 12 – log5 10 1.5440 – 1.4307 0.1133

  7. Properties of Logarithms For m > 0, n > 0, b > 0, and any real number p: Power Property logb mp = p logb m

  8. Example 3 given: log5 12  1.5440 log5 10  1.4307 log5 1254 5x = 125 = 4 log5 125 53 = 125 =4  3 x = 3 = 12

  9. Practice Write each expression as a single logarithm. 1) log2 14 – log2 7 2) log3 x + log3 4 – log3 2 3) 7 log3 y – 4 log3 x

  10. Homework p.382 #13-21 odds,31,35

  11. 4 minutes Warm-Up Write each expression as a single logarithm. Then simplify, if possible. 1) log6 6 + log6 30 – log6 5 2) log6 5x + 3(log6 x – log6 y)

  12. 6.4.2 Properties of Logarithmic Functions Objectives: Simplify and evaluate expressions involving logarithms Solve equations involving logarithms

  13. Properties of Logarithms For b > 0 and b  1: Exponential-Logarithmic Inverse Property logb bx = x and b logbx = x for x > 0

  14. Example 1 Evaluate each expression. a) b)

  15. Practice Evaluate each expression. 1) 7log711 – log3 81 2) log8 85 + 3log38

  16. Properties of Logarithms For b > 0 and b  1: One-to-One Property of Logarithms If logb x = logb y, then x = y

  17. Example 2 Solve log2(2x2 + 8x – 11) = log2(2x + 9) for x. log2(2x2 + 8x – 11) = log2(2x + 9) 2x2 + 8x – 11 = 2x + 9 2x2 + 6x – 20 = 0 2(x2 + 3x – 10) = 0 2(x – 2)(x + 5) = 0 x = -5,2 Check: log2(2x2 + 8x – 11) = log2(2x + 9) log2 (–1) = log2 (-1) undefined log2 13 = log2 13 true

  18. Practice Solve for x. 1) log5 (3x2 – 1) = log5 2x 2) logb (x2 – 2) + 2 logb 6 = logb 6x

  19. Homework p.382 #29,33,37,43,47,49,51,57,59,61

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