Warm-up3/13/08

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Warm-up 3/13/08

• Solve each equation.
• x2 169
• w3 216
• y3 -1/8
• z4 625

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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 its missing, youll be fined for it.
• Book Check

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• Copy SLM for Unit 6 (chapter 6)

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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?
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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
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Concept II
Radical Power Functions
Lesson Essential Question (LEQ)
How do you interpret graphs of rational power functions?
Vocabulary
Rational Power Function
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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
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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
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Concept V
Properties of Logarithms
Lesson Essential Question (LEQ)
How do you use the properties of logarithms to solve problems?
Vocabulary

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Concept VI
Solving Exponential Equations
Lesson Essential Question (LEQ)
How do you use properties of logarithms to solve exponential problems?
Vocabulary
Exponential Equation
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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 bgt0 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.

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Why?

• Why cant x 0?
• Any number raised to the 0 power is 1, and that
  would make it a constant function
• Why cant 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) v-4 2i

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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

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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)

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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.

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Nth Root Functions

• Functions with equations in the form y x1/n,
  where n is an integer gt 2 are called nth root
  functions
• X1/n is defined only when x0 (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

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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)

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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

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• Ex. y 5v32
• index (the root you want)
• y 321/5
• The principal root is the positive root.
• Ex. 4v81 9, -9 9 principal root
• Rational Exponent Property
• bm/n nvbm

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Simplify each expression

1. 82/3
2. 43/2
3. (53/4)4/3
4. 2434/5

1. 4
2. 8
3. 5
4. 81

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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 pr3

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Refresh on Properties

• x5x4
• x9
• (xy)2
• x2y2
• x6x4 x ? 0
• x2

• (x/y)5 x ? 0
• x5/y5
• x0
• 1
• X-n
• 1/(xn)

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Assignment

• Section 6.1
• p. 374 375
• 1 7, 10 12, 14, 16, 17 - 19

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Guided Practice

• 6.2 Worksheet

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Warm-up 3/17/08

• Rewrite each expression using a radical sign.
• x7/8
• x-8/5
• Evaluate without a calculator.
• 641/3
• 4) 16-3/4
• 5) 641/2

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Quick Review

• In textbooks,
• p.381
• Complete
• 15 17
• In notes

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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

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The relationship

• Exponential Equation
• y bx
• is equivalent to
• Logarithmic Equation
• logb(y) x
• Log-base-b of y equals x.

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

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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

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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

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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

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

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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.

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General Forms

• What is the value of logb1?
• 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.

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Solve

• 10x 4
• to undo power of 10, log
• log10x log4
• x log4
• x 0.60206

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Solve to the nearest tenth

• log x 3.724
• 103.724 x
• x 5296.6
• To check,
• Log (5296.6) 3.724?
• Yes!?

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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

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Practice

• p. 387 - 388 16 - 21
• 6.3 Worksheet
• Homework
• Section 6.3
• p.387-388
• 1 7, 9 - 15

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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 0.00005, what is the pH?

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Practice

• p. 387 - 388 16 - 21
• 6.3 Worksheet
• Homework
• Section 6.3
• p.387-388
• 1 7, 9 - 15

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Quiz

• 6.1 - 6.3
• 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.

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Warm-up 3/19/08

1. Use a calculator to give a 3-place decimal
   approximation to ln2 through ln10.
2. Which of them are sums of two other logarithms?

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Activity

• Class assignment
• p.389
• 1 - 3d
• Discuss

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Assignment

• Read p. 390 394
• Do 6.4 WS

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6.5 Properties of Logarithms

• LEQ How do you use properties of logarithms to
  simplify logarithmic problems?
• Recall the properties (p.398-400)

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Properties of Logarithms

• 1) logbMN logbM logbN
• Product Property
• logbM/N logbM logbN
• Quotient Property
• logbMk klogbM
• Power Property

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Examples

• Write the expression in single log form.
• log320 log34
• log320/4 log35
• 2) 3log2x log2y
• log2x3 log2y log2x3y
• 3log2 log4 log 16
• log(23 x 4)/16 log 32/16 log2

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Examples of expansion

• Expand each logarithm.
• log5(x/y)
• log5x log5y
• log3r4
• log3 logr4 log3 4 log r
• Can you expand log3(2x 1)?
• NO, the sum cant be factored.

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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

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Estimating Answers

• http//www.algebra.com/algebra/homework/logarithm/
  Properties-of-Logarithms.lesson

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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

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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

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Solve

• 2log9(vx) log9(6x 1) 0
• 2log9(vx) log9(6x 1)
• log9(vx)2 log9(6x 1)
• x 6x 1
• 1 5x
• 1/5 x
• Always plug the answer back in to make sure that
  it wont produce any negatives or zeros in the
  logarithms.

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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.

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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.

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• Sometimes its more useful to convert an
  equation to exponential form to work a problem.

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Solve

• log5(2x 4) 2
• 52 2x 4
• 25 2x 4
• 21 2x
• x 12.5
• 12.5 checks as a solution

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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, -2
• -2 would create a negative log,
• 5 works, so the only solution is 5.

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Solve

• Log2(x2 6x) 3 log2(1 x)
• Log2(x2 6x) - log2(1 x) 3
• Log2(x2 6x) 3
• (1 x)
• (x2 6x) 23
• (1 x)

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• (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)

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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. 4 x2/3 31

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Solve with reciprocal exponents

• 4 x3/2 31
• x3/2 (31 4)
• x3/2 27
• Multiply both sides by reciprocal of 3/2
• x(2/3)(3/2) 272/3
• x 27(1/3)2
• x 32
• x 9

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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

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

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Try these

1. 3x 4
2. 62x 21
3. 3x4 101

1. 1.262
2. 0.850
3. 0.201

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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

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Ex. of solving with change of base

• 62x 1500
• log662x log61500
• 2xlog66 log61500
• log66 1so
• 2x log1500/log6 (change of base)
• 2x 4.0816
• x 2.0408

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• Ex1) Use natural logs to solve 8e2x 20.
• 8e2x 20
• e2x 20/8
• e2x 2.5
• 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

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Assignment

• 6.5 WS
• Extra Practice?
• http//www-math.cudenver.edu/rbyrne/flash.htm
• http//www.purplemath.com/modules/index.htm

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Warm-up 7.6

• 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
• xmin0,xmax20ymin0ymax500yscl100
• Estimate when the 1st grade pop. will 250.
• When will the pop. reach 325?

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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

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Warm-up Test

1. You put 1500 into an account earning 7 interest
   compounded continuously. How long will it be
   until you have 2000 in your bank account?
2. Evaluate log4256
3. Expand log4r2t
4. Solve 3lnx ln2 4
5. Solve 2logx -4
6. Rewrite as a common log log316