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5-Minute Check on Activity 5-12

5-Minute Check on Activity 5-12. Using logarithmic regression from the following data, determine the function and the specified value to two decimal places: Function: F(18) =. y = a + b ln x y = 2089.97 + 630.49 ln x. y (18)= 2089.97 + 630.49 ln (18) = 3912.32.

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5-Minute Check on Activity 5-12

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  1. 5-Minute Check on Activity 5-12 Using logarithmic regression from the following data, determine the function and the specified value to two decimal places: Function: F(18) = y = a + b ln x y = 2089.97 + 630.49 ln x y (18)= 2089.97 + 630.49 ln (18) = 3912.32 Click the mouse button or press the Space Bar to display the answers.

  2. Activity 5 - 13 The Elastic Ball

  3. Objectives • Apply the log of a product property • Apply the log of a quotient property • Apply the log of a power property • Discover change of base formula

  4. Vocabulary • None new

  5. Activity You are continuing your work on the development of the elastic ball. You are still investigating the question, “If the ball is launched straight up, how far has it traveled vertically when it hits the ground for the 10th time?” However, your supervisor tells you that you cannot count the initial launch distance. You must calculate only the rebound distance. Using some physical properties, timers and your calculator, you collect the following data.

  6. Activity Cont Plot the data using STATPLOT What does the graph say about a model: linear, exponential or logarithmic? The data can be modeled (thru logarithmic regression) as T = 26.75 log N Plot both and see if the model is reasonable logarithmic

  7. Activity Cont Plots

  8. Properties of Logarithms logb (A×B) = logb A + logb B Product Property: Division Property: Exponent Property: Change of Base Property: No Sum or Subtraction Property: logb (A÷B) = logb A – logb B logb (Ap) = p logb A loga x logb (x) = ----------- loga b logb (A ± B) ≠ logb A ± logb B PLEASE NOTE!

  9. Product Examples log2 (2×2×2×2×2) = log2 2 + log2 2 + log2 2 + log2 2 + log2 2 = 5 log2 2 = 5 Log2 32 = Log (5st) = Log4 3 + log4 9 = Log2 7 + log2 11 = log 5 + log s + log t log4 (3×9) = log4 27 log2 (7×11) = log2 77

  10. Quotient Examples log2 (2) + log2 (x) – log2 (y) = 1 + log2 (x) – log2 (y) Log2 (2x/y) = Log (¾st) = Log4 3 - log4 9 = Log2 7 - log2 11 = log 3 – log 4 + log s + log t log4 (3÷9) = log4 (1/3) log2 (7÷11) = log2 (7/11)

  11. Power Examples log2 (25) = 5 log2 2 = 5 Log2 32 = Ln (st)7 = Log4 √51 = 4Log2 3 = 7(ln s + ln t) ½ log4 (3×17) = ½ (log4 3 + log4 3) log2 (34) = log2 81

  12. Change of Base Examples Most calculators have only log (base 10) and ln (base e) functions defined. In order to use logs to other bases, we need to convert to one of these two. Log3 (5) = Log4 x = Log2 7x = ln 5 ----------- ≈ 1.465 ln 3 ln x -------- ≈ 0.72135 ln x ln 4 ln 7x -------- ≈ 1.4427 (ln 7 + ln x) ln 2 ≈ 2.8074 + 1.4427 ln x

  13. Summary and Homework • Summary Properties of Logarithmic Functions: • logb (A ∙ B) = logb A + logb B • logb (A / B) = logb A – logb B • logb (A)p = p∙logb A • Change bases: • Remember: logb (x + y) ≠ logb x + logb y • Homework • pg 662-665; problems 1, 2, 6-9 log x ln x logb x = ------- or logb x = ------ log b ln b

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