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5.7 – Exponential Equations

5.7 – Exponential Equations. 5.7 Exponential Equations. Objectives: Solve Exponential Equations using the Change of Base Formula Evaluate logarithms Solve logarithms Vocabulary: logarithms, natural logarithms. Daily Objectives. Perform the Change of Base formula.

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5.7 – Exponential Equations

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  1. 5.7 – Exponential Equations

  2. 5.7 Exponential Equations Objectives: • Solve Exponential Equations using the Change of Base Formula • Evaluate logarithms • Solve logarithms Vocabulary: logarithms, natural logarithms

  3. Daily Objectives • Perform the Change of Base formula. • Master solving tricky logarithm equations. • Exponent variables

  4. Topic One: Change of Base Formula The BASE goes to the BOTTOM

  5. Example 1: Using Change of Base Formula On the other hand, why is problem #2 not the type of problem that you should use the change of base formula on? • = 3 • log 1000 This is a great problems to use the change of base formula on – Why? #3 can work with either – Why?

  6. Example 2: Solve for a variable exponent

  7. Example 3: Solve for a variable exponent

  8. Example 4: Solve for a variable exponent

  9. Example 5: Solve for a variable exponent

  10. Solving Logarithmic Equations Example 14: Newton’s Law of Cooling • The temperature T of a cooling substance at time t (in minutes) is: • T = (T0 – TR) e-rt + TR • T0= initial temperature • TR= room temperature • r = constant cooling rate of the substance

  11. Solving Logarithmic Equations Example 14: You’re cooking stew. When you take it off the stove the temp. is 212°F. The room temp. is 70°F and the cooling rate of the stew is r = 0.046. How long will it take to cool the stew to a serving temp. of 100°? T = (T0 – TR) e -rt + TR T0 = 212, TR = 70, T = 100 r = 0.046 So solve: 100 = (212 – 70)e -0.046t + 70

  12. Solving Logarithmic Equations 30 = 142e -0.046t(subtract 70) 15/71 = e -0.046t(divide by 142) • How do you get the variable out of the exponent? ln(15/71) = lne-.046t (take the ln of both sides) ln(15/71) = – 0.046t ln(15/71)/(– 0.046) = t t =(ln15 – ln71)/(– 0.046) = t ≈ – 1.556/(– 0.046) t ≈ 33.8 about 34 minutes to cool!

  13. Homework • TB p. 205 #9-14, 23

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