5-Minute Check on Activity 5-125-Minute Check on Activity 5-125-Minute Check on Activity 5-125-Minute Check on Activity 5-12

Click the mouse button or press the Space Bar to display the answers.Click the mouse button or press the Space Bar to display the answers.

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

X 1 2 3 4 5 6

Y 2201 2422 2688 2902 3144 3331

y (18)= 2089.97 + 630.49 ln (18) = 3912.32

Activity 5 - 13

The Elastic Ball

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

Vocabulary• None new

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.

Times Ball hits Ground 1 2 3 4 5 6

Total Rebound Distance 0 9.0 13.5 16.3 18.7 21.0

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

Times Ball hits Ground 1 2 3 4 5 6

Total Rebound Distance 0 9.0 13.5 16.3 18.7 21.0

logarithmic

Activity Cont

Plots

Times Ball hits Ground 1 2 3 4 5 6

Total Rebound Distance 0 9.0 13.5 16.3 18.7 21.0

Properties of Logarithms

Product Property:

Division Property:

Exponent Property:

Change of Base Property:

No Sum or Subtraction Property:

logb (A×B) = logb A + logb B

logb (A÷B) = logb A – logb B

logb (Ap) = p logb A

loga xlogb (x) = ----------- loga b

logb (A ± B) ≠ logb A ± logb B

PLEASE NOTE!

Product Examples

Log2 32 =

Log (5st) =

Log4 3 + log4 9 =

Log2 7 + log2 11 =

log2 (2×2×2×2×2) = log2 2 + log2 2 + log2 2 + log2 2 + log2 2= 5 log2 2 = 5

log4 (3×9) = log4 27

log2 (7×11) = log2 77

log 5 + log s + log t

Quotient Examples

Log2 (2x/y) =

Log (¾st) =

Log4 3 - log4 9 =

Log2 7 - log2 11 =

log2 (2) + log2 (x) – log2 (y) = 1 + log2 (x) – log2 (y)

log4 (3÷9) = log4 (1/3)

log2 (7÷11) = log2 (7/11)

log 3 – log 4 + log s + log t

Power Examples

Log2 32 =

Ln (st)7 =

Log4 √51 =

4Log2 3 =

log2 (25) = 5 log2 2 = 5

½ log4 (3×17) = ½ (log4 3 + log4 3)

log2 (34) = log2 81

7(ln s + ln t)

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 x-------- ≈ 0.72135 ln x ln 4

ln 7x-------- ≈ 1.4427 (ln 7 + ln x) ln 2

≈ 2.8074 + 1.4427 ln x

ln 5----------- ≈ 1.465 ln 3

Summary and Homework

• SummaryProperties 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 xlogb x = ------- or logb x = ------ log b ln b