1. Warm-up 3/13/08 Solve each equation. x2 = 169 w3= 216 y3 = -1/8
z4 = 625
12a8
4a3
-18m3n3 X7
1/p4q2
s7t4

2. Today is the last day for any make ups in the first 9 weeks
Heads Up!
Today is the last day for any make ups in the first 9 weeks
If you “borrowed” any calculators, please return them; I will be checking all of them by their serial numbers this afternoon, if you checked one out and it’s missing, you’ll be fined for it.
Book Check

3. Copy SLM for Unit 6 (chapter 6)

4. Topic:
Root, Power, and Logarithm Functions
Key Learning(s):
Use rational exponents to model situations
Solve problems arising from exponential or logarithmic models
Unit Essential Question (UEQ):
How do you use common and natural logarithms to model real life situations and solve problems?

5. Lesson Essential Question (LEQ):
Concept I:
Nth Root Functions
Lesson Essential Question (LEQ):
How do you use rational exponents to model situations?
Vocabulary:
Square root, cube root, nth root, radical

6. Radical Power Functions Lesson Essential Question (LEQ):
Concept II
Radical Power Functions
Lesson Essential Question (LEQ):
How do you interpret graphs of rational power functions?
Vocabulary:
Rational Power Function

7. Lesson Essential Question (LEQ):
Concept III
Logarithm Functions
Lesson Essential Question (LEQ):
How do you graph and evaluate logarithmic functions?
Vocabulary:
Exponential growth function, exponential growth curve, exponential decay curve, strictly increasing, asymptote, strictly decreasing

8. e and natural logarithms Lesson Essential Question (LEQ):
Concept IV
e and natural logarithms
Lesson Essential Question (LEQ):
How do you solve problems arising from exponential or logarithmic problems?
Vocabulary:
Exponential function with base e
Natural logarithm function

9. Properties of Logarithms Lesson Essential Question (LEQ):
Concept V
Properties of Logarithms
Lesson Essential Question (LEQ):
How do you use the properties of logarithms to solve problems?
Vocabulary:

10. Solving Exponential Equations Lesson Essential Question (LEQ):
Concept VI
Solving Exponential Equations
Lesson Essential Question (LEQ):
How do you use properties of logarithms to solve exponential problems?
Vocabulary:
Exponential Equation

11. §6.1: nth Root Functions
LEQ: How do you use rational roots to model situations? Defining the exponential function If b is any number such that b>0 and b≠0 then an exponential function is a function in the form, f(x)=bx b is the base x can be any real number.

12. Why? Why can’t x = 0? Why can’t x be negative?
Any number raised to the 0 power is 1, and that would make it a constant function
Why can’t x be negative?
If x was negative and it was raised to a fraction power, it would make the answer imaginary.
Ex) -4(1/2) = √-4 = 2i

13. Powers and Roots If x2 = k X is a square root of k If x3 = k
X is a cube root of k
If xn = k
X is an nth root of k

14. How many real roots are there?
When “n” is odd,
K has exactly one real nth root
When “n” is even and k is positive
Ex) x2n = k
K has two real nth roots
When “n” is even and k is negative
Ex) x2n = -k
K has no real nth roots (imaginary roots)

15. Nth root, nth power
Taking the nth root and the nth power of a number are inverse operations.
Ex) 104 = (10000)1/4
In general:
F(x) = xn
G(x) = x1/n
Are inverses of each other.

16. Nth Root Functions
Functions with equations in the form y = x1/n, where n is an integer > 2 are called nth root functions
X1/n is defined only when x≥0 (in this chapter)
The domain of these functions is all positive real numbers
The range is also the set of all positive real numbers.
Graph some: y = x1/2; y = x1/3; y = x1/4

17. Common Characteristics
What are some common characteristics of nth root function graphs?
All of them start at (0,0)
All of them pass through (1,0)

18. Inverse Refresh… To undo a square, you square root.
To unto a cube, you cube root.
To undo an exponent of 4, you take the 4th root.
You can use a radical symbol or a rational exponent to indicate roots.
*Different ways to use calculators…

19. Ex. y = 5√32
index (the root you want)
y = 321/5
The principal root is the positive root.
Ex. 4√81 = +9, principal root
Rational Exponent Property
bm/n = n√bm

20. Simplify each expression
82/3
43/2
(53/4)4/3
2434/5 4 8 5 81

21. Solving in terms of variables
Express the radius r in mm of a spherical ball bearing as a function of the volume V in mm3. (Solve the equation for r) V = 4/3 πr3

22. Refresh on Properties x5x4 x9 (xy)2 x2y2 x6÷x4 x ≠ 0 x2
(x/y)5 x ≠ 0 x5/y5 x0 1 X-n 1/(xn)

23. Assignment
Section 6.1 p. 374 – 375 #1 – 7, 10 – 12, 14, 16,

24. Guided Practice
6.2 Worksheet

25. Warm-up 3/17/08 Rewrite each expression using a radical sign. x7/8
Evaluate without a calculator.
641/3
4) 16-3/4
5) 641/2

26. In textbooks, p.381 Complete #15 – 17 In notes
Quick Review
In textbooks, p.381 Complete #15 – 17 In notes

27. §6.3: Converting Exponentials
LEQ: How do you convert between exponential and logarithmic forms? How do you solve problems arising from exponential or logarithmic forms? Logarithms are the “opposite” of exponents; they “undo” exponentials. Logarithms have a specific relationship to exponentials…

28. The relationship Exponential Equation: y = bx is equivalent to:
Logarithmic Equation:
logb(y) = x
“Log-base-b of y equals x.”
B is the base of the logarithm just as b is the base in b^x.
Just as the base b is positive and not equal to 1, so also the logarithmic base b is positive and not equal to 1.

29. Quick Summary Equation Meaning Base(exponent) = power
Exponent = log of power, base b
bt = p
t = logbp
102 = 100
2 = log10(100)

30. Ex. Convert 63 = 216 to the equivalent logarithmic form.
Base is 6, exponent is 3…
log6(216) = 3
Ex. Convert log4(1024) = 5 to the equivalent exponential expression.
Base is 4, exponent is 5
45 = 1024

31. Exponent in a different form
log443= 3
43 = 43
log552 = x
5x = 52
log225 = y
2y = 25
y = 5
logbb2 = z
bz = b2
z = 2

32. To evaluate logarithms, you can write them in exponential form…
From Logs to Exponents
52 = 25
base is 5, exp is 2: log5(25) = 2
(1/2)3 = 1/8
log(1/2)(1/8) = 3
24 = 64
log2(64) = 4
To evaluate logarithms, you can write them in exponential form…

33. Evaluating Exponents Ex. Log816 8x = 16
Both can be written with a base of 2:
23x = 24
Now you can set the exponents equal
(since the bases are equal)
3x = 4 Solve for x
X = 4/3
So, log816 = 4/3

34. Since both bases are five, you can set the exponents equal.
Evaluate log5125.
5x = 125
5x = 53
Since both bases are five, you can set the exponents equal.
X = 3
Thus, log5125 = 3

35. A short cut To evaluate log216, you can ask yourself
“What power of 2 is equal to 16”.
What question would you ask to evaluate log327? Evaluate it.
What question would you ask to evaluate log10100? Evaluate it.

36. General Forms What is the value of logb1? What is the value of logbb?
Log21 c) Log31
Log41 d) Log51
What is the value of logbb?
log22 c) log44
Log55 d) log99
Explain why the base b in y = logbx cannot equal 1.
Log(b)1 = 0
Log(b)b = 1

37. 10x = 4 to “undo” power of 10, log… log10x = log4 x = log4 x = 0.60206
Solve
10x = 4 to “undo” power of 10, log… log10x = log4 x = log4 x =

38. Solve to the nearest tenth
log x = = x x ≈ To check, Log (5296.6) = 3.724? Yes!

39. Inverse Graphs
Logarithms and exponentials are also inverses of each other.
If you graph y = 10x and y = logx, they are inverses. Their graphs are reflections of each other over the line y = x.
*log has a general base of 10*
Find the inverse function of each:
Y = 3x 2) y = 5x 3) y = bx
1) Y = log3x 2) y = log5x 3) y = logbx

40. Practice
p # Worksheet Homework Section 6.3 p #1 – 7,

41. Warm-up 3/18/08
Consider two spheres, one with a radius of 2cm and the other with twice the volume of the first.
Find the volume of the larger sphere.
Find the radius of the larger sphere.
The concentration of a hydrogen ions in a aqueous solution is given by the formula [H+] = 10(-pH)
Find the concentration if the pH is 1.5
If the concentration is , what is the pH?

42. Practice
p # Worksheet Homework Section 6.3 p #1 – 7,

43. Quiz
Group Quiz (You may use a partner, notes, book, etc, but keep in mind you have a test coming up soon.) It may be a good idea to use each other to “check” your work.

44. Warm-up 3/19/08
Use a calculator to give a 3-place decimal approximation to ln2 through ln10.
Which of them are sums of two other logarithms?

45. Class assignment p.389 #1 - 3d Discuss
Activity
Class assignment p.389 #1 - 3d Discuss

46. Assignment
Read p. 390 – 394 Do: 6.4 WS

47. §6.5: Properties of Logarithms
LEQ: How do you use properties of logarithms to simplify logarithmic problems? “Recall” the properties (p )

48. Properties of Logarithms
1) logbMN = logbM + logbN
Product Property
logbM/N = logbM – logbN
Quotient Property
logbMk= klogbM
Power Property

49. Examples Write the expression in single log form. log320 – log34
2) 3log2x + log2y
log2x3 + log2y = log2x3y
3log2 + log4 – log 16
log(23 x 4)/16 = log 32/16 = log2

50. Examples of expansion Expand each logarithm. log5(x/y) log5x – log5y
log3r4
log3 + logr4 = log log r
Can you expand log3(2x + 1)?
NO, the sum can’t be factored.

51. Warm-up 3/21/08 Rewrite each expression logx + logy 2logx – logx
log3 + log4
Evaluate the expression
4) log28 – log22
log(xy)
logx
log12 2

52. Estimating Answers

53. Evaluating Expressions
To evaluate an expression, apply rules, then use the logarithmic rules to solve.
Ex. log55 – log5125
log5(5/125)
log5(1/25)
log5(1/52)
5x =5-2
x = -2

54. Solving Logarithm Tips:
Logarithms should contain the same base (or you have to use a formula to change them)
You can only plug positive numbers into a logarithm
If you have two logs in a problem, one on each side of the equal sign and both with a coefficient of one, you can “drop” the logarithms

55. Solve
2log9(√x) – log9(6x – 1) = 0 2log9(√x) = log9(6x – 1) log9(√x)2= log9(6x – 1) x= 6x – 1 1 = 5x 1/5 = x Always plug the answer back in to make sure that it won’t produce any negatives or zeros in the logarithms.

56. Solve
logx + log(x – 1) = log(3x + 12) log(x(x – 1)) = log(3x + 12) x2 – x = 3x + 12 x2 – 4x - 12 = 0 (x – 6)(x + 2) = 0 X = 6, -2 When you plug them back in, 6 works, but -2 is an extraneous solution.

57. Solve
ln10 – ln(7 – x) = lnx ln10/(7 – x) = lnx 10 = x 7 – x 10 = x(7 – x) 10 = 7x – x2 x2 – 7x + 10 = 0 (x – 5)(x – 2) = 0 x = 5, 2 Both are solutions.

58. Sometimes it’s more useful to convert an equation to exponential form to work a problem.

59. Solve
log5(2x + 4) = 2 52 = 2x = 2x = 2x x = checks as a solution

60. Solve
logx = 1 – log(x – 3) logx + log(x – 3) = 1 log(x(x – 3) = 1 log(x2 – 3x) = 1 x2 – 3x = 101 x2 – 3x – 10 = 0 (x – 5)(x + 2) = 0 x = 5, would create a negative log, 5 works, so the only solution is 5.

61. Solve
Log2(x2 – 6x) = 3 + log2(1 – x) Log2(x2 – 6x) - log2(1 – x) = 3 Log2(x2 – 6x) = 3 (1 – x) (x2 – 6x) = 23

62. (x2 – 6x) = 8 (1 – x) (x2 – 6x) = 8(1 – x) x2 – 6x = 8 – 8x x2 – 6x +8x = 8 x2 +2x - 8 = 0 (x + 4)(x – 2) = 0 x = -4, 2 The only solution that works in the problem is -4 (2 is extraneous)

63. Radical Equation
To solve equations in the form xa = c, where the variable is raised to a power, you can either use the properties of exponents or use radicals.
Radical Equation –
The variable in an equation occurs in a radicand.
Ex x2/3 = 31

64. Solve with reciprocal exponents:
Multiply both sides by reciprocal of 3/2
x(2/3)(3/2) = 272/3
x = 27(1/3)2
x = 32
x = 9

65. Exponential Equations
An exponential equation can be solved by taking the logarithm of both sides and then using the power rule to simplify the problems.
Ex) (3)x = 36
log(3)x = log36
xlog(3) = log36
x = log36/log3
x = 3.26

66. Use rules of exponents to get: 3xlog7 = log20
Ex. 73x = 20
log73x = log20
Use rules of exponents to get:
3xlog7 = log20
Divide both sides by 3log7
x = log20/3log7
x = 0.513

67. Try these
3x = 4
62x = 21
3x+4 = 101
1.262
0.850
0.201

68. What about other bases?
To evaluate a logarithm with any base, you can use the change of base formula.
*for this formula, the bases cannot = 1.
logbM = log10M
log10b
Ex. Log35 = log5/log3

69. Ex. of solving with change of base
log662x = log61500
2xlog66 = log61500
log66 = 1…so
2x = log1500/log6 (change of base)
2x =
x =

70. Ex1) Use natural logs to solve 8e2x= 20. 8e2x = 20 e2x = 20/8
lne2x = ln2.5
2xlne = ln2.5
*ln(e) cancel each other out just like log(10)
2x = ln2.5
X = ln2.5/2
*Make sure you close parenthesis around top!
X = 0.458

71. Assignment 6.5 WS Extra Practice?

72. Warm-up
The function y = 200(1.04)x models the first grade population y of an elementary school x years after the year 2000.
Graph the function on your graphing calculator. Adjust the viewing window:
xmin=0,xmax=20;ymin=0;ymax=500;yscl=100
Estimate when the 1st grade pop. will = 250.
When will the pop. reach 325? 1)
About 2006

73. Warm-up 7.5 What are the 3 properties of logarithms?
Why do we need those properties?
Simplify
41/2
272/3
(1/9)-1/2
163/4
quotient, product, power; you can re-write logarithmic functions as power function and make them more simple to evaluate in a calculator. 2 9 3 8

74. Warm-up Test
You put $1500 into an account earning 7% interest compounded continuously. How long will it be until you have $2000 in your bank account?
Evaluate: log4256
Expand: log4r2t
Solve: 3lnx – ln2 = 4
Solve: 2logx = -4
Rewrite as a common log: log316
4.1 yr 4
2log4r + log4t
4.78
0.01
Log16/log3