Unit 2 Logarithms 10-11-12. DO NOW Expand the logarithm and simplify if possible Log 5 3 2 x Answer:...
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Transcript of Unit 2 Logarithms 10-11-12. DO NOW Expand the logarithm and simplify if possible Log 5 3 2 x Answer:...
![Page 1: Unit 2 Logarithms 10-11-12. DO NOW Expand the logarithm and simplify if possible Log 5 3 2 x Answer: 2 log 5 3 – log 5 x.](https://reader035.fdocuments.us/reader035/viewer/2022081506/56649de85503460f94ae22bc/html5/thumbnails/1.jpg)
Unit 2 Logarithms
10-11-12
![Page 2: Unit 2 Logarithms 10-11-12. DO NOW Expand the logarithm and simplify if possible Log 5 3 2 x Answer: 2 log 5 3 – log 5 x.](https://reader035.fdocuments.us/reader035/viewer/2022081506/56649de85503460f94ae22bc/html5/thumbnails/2.jpg)
DO NOW
• Expand the logarithm and simplify if possible• Log 5 3
2
x
• Answer: 2 log 5 3 – log 5 x
![Page 3: Unit 2 Logarithms 10-11-12. DO NOW Expand the logarithm and simplify if possible Log 5 3 2 x Answer: 2 log 5 3 – log 5 x.](https://reader035.fdocuments.us/reader035/viewer/2022081506/56649de85503460f94ae22bc/html5/thumbnails/3.jpg)
activating
• http://youtu.be/zzu2POfYv0Y?t=1s
![Page 4: Unit 2 Logarithms 10-11-12. DO NOW Expand the logarithm and simplify if possible Log 5 3 2 x Answer: 2 log 5 3 – log 5 x.](https://reader035.fdocuments.us/reader035/viewer/2022081506/56649de85503460f94ae22bc/html5/thumbnails/4.jpg)
7 – 4
• Objective: Understand the properties of logarithms
• Objective: expand and condense logarithmic expressions
• Objective: Change of base formula
![Page 5: Unit 2 Logarithms 10-11-12. DO NOW Expand the logarithm and simplify if possible Log 5 3 2 x Answer: 2 log 5 3 – log 5 x.](https://reader035.fdocuments.us/reader035/viewer/2022081506/56649de85503460f94ae22bc/html5/thumbnails/5.jpg)
examples
1. Reasoning: Can you expand log 3 (2x + 1) ? Explain
• No, the expression (2x + 1) is a sum, so it is not covered by the product, quotient, or power properties
2. Write the logarithmic expression as a single logarithm :1/2 ( log
x 4 + log x y) – 3 log x z
• Log x 2√ y
z 3
![Page 6: Unit 2 Logarithms 10-11-12. DO NOW Expand the logarithm and simplify if possible Log 5 3 2 x Answer: 2 log 5 3 – log 5 x.](https://reader035.fdocuments.us/reader035/viewer/2022081506/56649de85503460f94ae22bc/html5/thumbnails/6.jpg)
More examples for you to try
• Write each logarithm as the quotient of 2 common logarithms. Do not simplify the quotient
Pg. 467 #68, 69(hint log answer/log base)• Evaluate each logarithm– Pg. 468 # 93– Pg 468 # 54
![Page 7: Unit 2 Logarithms 10-11-12. DO NOW Expand the logarithm and simplify if possible Log 5 3 2 x Answer: 2 log 5 3 – log 5 x.](https://reader035.fdocuments.us/reader035/viewer/2022081506/56649de85503460f94ae22bc/html5/thumbnails/7.jpg)
One more problem
• What is the value of log 7 25? Use the change of base formula
• About 1.65
![Page 8: Unit 2 Logarithms 10-11-12. DO NOW Expand the logarithm and simplify if possible Log 5 3 2 x Answer: 2 log 5 3 – log 5 x.](https://reader035.fdocuments.us/reader035/viewer/2022081506/56649de85503460f94ae22bc/html5/thumbnails/8.jpg)
worksheets
• 7 -4 think about a plan• 7 -4 puzzle: letter scramble
![Page 9: Unit 2 Logarithms 10-11-12. DO NOW Expand the logarithm and simplify if possible Log 5 3 2 x Answer: 2 log 5 3 – log 5 x.](https://reader035.fdocuments.us/reader035/viewer/2022081506/56649de85503460f94ae22bc/html5/thumbnails/9.jpg)
7-5, 7-6, and polynomials
10-15-12
![Page 10: Unit 2 Logarithms 10-11-12. DO NOW Expand the logarithm and simplify if possible Log 5 3 2 x Answer: 2 log 5 3 – log 5 x.](https://reader035.fdocuments.us/reader035/viewer/2022081506/56649de85503460f94ae22bc/html5/thumbnails/10.jpg)
Do now
• Pg. 461 #22• Pg. 467 #70 and 72• Pg. 473 # 7
![Page 11: Unit 2 Logarithms 10-11-12. DO NOW Expand the logarithm and simplify if possible Log 5 3 2 x Answer: 2 log 5 3 – log 5 x.](https://reader035.fdocuments.us/reader035/viewer/2022081506/56649de85503460f94ae22bc/html5/thumbnails/11.jpg)
7 - 5
• How can you solve exponential equations? • Objective: solve logarithmic equations using
technology and algebraically
![Page 12: Unit 2 Logarithms 10-11-12. DO NOW Expand the logarithm and simplify if possible Log 5 3 2 x Answer: 2 log 5 3 – log 5 x.](https://reader035.fdocuments.us/reader035/viewer/2022081506/56649de85503460f94ae22bc/html5/thumbnails/12.jpg)
Exponential equation
• Any equation that contains the form bcx, as a = bcx, where the exponent includes a variable
- Remember, you can use LOGARITHMS to solve exponential equations
- You can use EXPONENTS to solve logarithmic equations
![Page 13: Unit 2 Logarithms 10-11-12. DO NOW Expand the logarithm and simplify if possible Log 5 3 2 x Answer: 2 log 5 3 – log 5 x.](https://reader035.fdocuments.us/reader035/viewer/2022081506/56649de85503460f94ae22bc/html5/thumbnails/13.jpg)
examples
• Solving an exponential equation – common base
• Pg. 469• Finding solutions• Use power property of exponents to solve
![Page 14: Unit 2 Logarithms 10-11-12. DO NOW Expand the logarithm and simplify if possible Log 5 3 2 x Answer: 2 log 5 3 – log 5 x.](https://reader035.fdocuments.us/reader035/viewer/2022081506/56649de85503460f94ae22bc/html5/thumbnails/14.jpg)
examples
• Solving an exponential equation – different bases
• Finding solutions• Solve by taking logarithm of each side of the
equation
![Page 15: Unit 2 Logarithms 10-11-12. DO NOW Expand the logarithm and simplify if possible Log 5 3 2 x Answer: 2 log 5 3 – log 5 x.](https://reader035.fdocuments.us/reader035/viewer/2022081506/56649de85503460f94ae22bc/html5/thumbnails/15.jpg)
Solving an exponential equation with graph or table
![Page 16: Unit 2 Logarithms 10-11-12. DO NOW Expand the logarithm and simplify if possible Log 5 3 2 x Answer: 2 log 5 3 – log 5 x.](https://reader035.fdocuments.us/reader035/viewer/2022081506/56649de85503460f94ae22bc/html5/thumbnails/16.jpg)
Modeling with exponential equations
• Logarithmic equation: is an equation that includes one or more logarithms involving a variable
• Pg. 477
![Page 17: Unit 2 Logarithms 10-11-12. DO NOW Expand the logarithm and simplify if possible Log 5 3 2 x Answer: 2 log 5 3 – log 5 x.](https://reader035.fdocuments.us/reader035/viewer/2022081506/56649de85503460f94ae22bc/html5/thumbnails/17.jpg)
Using logarithmic properties to solve an equation
![Page 18: Unit 2 Logarithms 10-11-12. DO NOW Expand the logarithm and simplify if possible Log 5 3 2 x Answer: 2 log 5 3 – log 5 x.](https://reader035.fdocuments.us/reader035/viewer/2022081506/56649de85503460f94ae22bc/html5/thumbnails/18.jpg)
Solving a logarithmic equation
• Problems in book and worksheet
![Page 19: Unit 2 Logarithms 10-11-12. DO NOW Expand the logarithm and simplify if possible Log 5 3 2 x Answer: 2 log 5 3 – log 5 x.](https://reader035.fdocuments.us/reader035/viewer/2022081506/56649de85503460f94ae22bc/html5/thumbnails/19.jpg)
H.O.T. question/activity/task
• Given y = ab cx
• Explain how replacing c with ( - c ) affects the function
![Page 20: Unit 2 Logarithms 10-11-12. DO NOW Expand the logarithm and simplify if possible Log 5 3 2 x Answer: 2 log 5 3 – log 5 x.](https://reader035.fdocuments.us/reader035/viewer/2022081506/56649de85503460f94ae22bc/html5/thumbnails/20.jpg)
Wrap up
• How are logarithms and exponential functions related to real-world data? (actual events, weather, money, etc). In your answer identify behaviors that tend to be explained using logarithmic and exponential functions (use the terms learned)
• Answers: radioactivity, hurricanes, population growth, stock market, compound interest
![Page 21: Unit 2 Logarithms 10-11-12. DO NOW Expand the logarithm and simplify if possible Log 5 3 2 x Answer: 2 log 5 3 – log 5 x.](https://reader035.fdocuments.us/reader035/viewer/2022081506/56649de85503460f94ae22bc/html5/thumbnails/21.jpg)
Doubling time discovery
• Test thursday unit 2, complete or try to finish review packet
• http://www.regentsprep.org/Regents/math/ALGEBRA/AE7/ExpDecayL.htm
![Page 22: Unit 2 Logarithms 10-11-12. DO NOW Expand the logarithm and simplify if possible Log 5 3 2 x Answer: 2 log 5 3 – log 5 x.](https://reader035.fdocuments.us/reader035/viewer/2022081506/56649de85503460f94ae22bc/html5/thumbnails/22.jpg)
7-6 • Natural logarithms pg. 478• The function y = ex has an inverse, the natural
logarithmic function, y = logex, or y = ln x
• Y = ex and y = ln x are inverse functions• a = eb then b = ln a
![Page 23: Unit 2 Logarithms 10-11-12. DO NOW Expand the logarithm and simplify if possible Log 5 3 2 x Answer: 2 log 5 3 – log 5 x.](https://reader035.fdocuments.us/reader035/viewer/2022081506/56649de85503460f94ae22bc/html5/thumbnails/23.jpg)
Log vs LN• Sometimes it is easier to think of logs in these terms instead! So, the
difference is in the base -- ln has base e, log has base 10. • The log button on your calculator is known as the common logarithm
which is of base 10. The ln button on your calculator has a base of "e". Here is what they look like:Base 10 y = log(10) xNatural Base y = log(e)xWritten as y = ln xThere are a couple of reasons why we use the natural logarithm versus the logarithm of base, b. When dealing with log, there are 2 variables that can affect the function, the base and the x value. With ln, since the base is always "e", the only factor affecting the function is x. It just makes it easier to manipulate and use mathematically.
![Page 24: Unit 2 Logarithms 10-11-12. DO NOW Expand the logarithm and simplify if possible Log 5 3 2 x Answer: 2 log 5 3 – log 5 x.](https://reader035.fdocuments.us/reader035/viewer/2022081506/56649de85503460f94ae22bc/html5/thumbnails/24.jpg)
http://www.wikihow.com/Understand-Logarithms
![Page 25: Unit 2 Logarithms 10-11-12. DO NOW Expand the logarithm and simplify if possible Log 5 3 2 x Answer: 2 log 5 3 – log 5 x.](https://reader035.fdocuments.us/reader035/viewer/2022081506/56649de85503460f94ae22bc/html5/thumbnails/25.jpg)
Begin unit 3
• Unit 3 diagnostic test• Homework if don’t get too
![Page 26: Unit 2 Logarithms 10-11-12. DO NOW Expand the logarithm and simplify if possible Log 5 3 2 x Answer: 2 log 5 3 – log 5 x.](https://reader035.fdocuments.us/reader035/viewer/2022081506/56649de85503460f94ae22bc/html5/thumbnails/26.jpg)
Unit 3 Polynomials
![Page 27: Unit 2 Logarithms 10-11-12. DO NOW Expand the logarithm and simplify if possible Log 5 3 2 x Answer: 2 log 5 3 – log 5 x.](https://reader035.fdocuments.us/reader035/viewer/2022081506/56649de85503460f94ae22bc/html5/thumbnails/27.jpg)
Agenda
1. Do now2. Activating3. What’s up next?
![Page 28: Unit 2 Logarithms 10-11-12. DO NOW Expand the logarithm and simplify if possible Log 5 3 2 x Answer: 2 log 5 3 – log 5 x.](https://reader035.fdocuments.us/reader035/viewer/2022081506/56649de85503460f94ae22bc/html5/thumbnails/28.jpg)
What you will be able to do?
• Factor polynomials• Describe end behavior of polynomials• Find the inverse of functions• Know and apply the binomial theorem• Recognize a polynomial function in real-world
situation• What does the degree of a polynomial tell you
about its related polynomial function?
![Page 29: Unit 2 Logarithms 10-11-12. DO NOW Expand the logarithm and simplify if possible Log 5 3 2 x Answer: 2 log 5 3 – log 5 x.](https://reader035.fdocuments.us/reader035/viewer/2022081506/56649de85503460f94ae22bc/html5/thumbnails/29.jpg)
Vocabulary
• Zeroes• Binomial expansion• Multiplicity• Relative extrema• concavity
![Page 30: Unit 2 Logarithms 10-11-12. DO NOW Expand the logarithm and simplify if possible Log 5 3 2 x Answer: 2 log 5 3 – log 5 x.](https://reader035.fdocuments.us/reader035/viewer/2022081506/56649de85503460f94ae22bc/html5/thumbnails/30.jpg)
Operations of polynomials
• Add, subtract, multiply polynomials• Synthetic division• Remainder theorem
![Page 31: Unit 2 Logarithms 10-11-12. DO NOW Expand the logarithm and simplify if possible Log 5 3 2 x Answer: 2 log 5 3 – log 5 x.](https://reader035.fdocuments.us/reader035/viewer/2022081506/56649de85503460f94ae22bc/html5/thumbnails/31.jpg)
Operations of polynomial problems
![Page 32: Unit 2 Logarithms 10-11-12. DO NOW Expand the logarithm and simplify if possible Log 5 3 2 x Answer: 2 log 5 3 – log 5 x.](https://reader035.fdocuments.us/reader035/viewer/2022081506/56649de85503460f94ae22bc/html5/thumbnails/32.jpg)
Binomial expansion
• Pascal’s triangle
![Page 33: Unit 2 Logarithms 10-11-12. DO NOW Expand the logarithm and simplify if possible Log 5 3 2 x Answer: 2 log 5 3 – log 5 x.](https://reader035.fdocuments.us/reader035/viewer/2022081506/56649de85503460f94ae22bc/html5/thumbnails/33.jpg)
Binomial
• Examples:
![Page 34: Unit 2 Logarithms 10-11-12. DO NOW Expand the logarithm and simplify if possible Log 5 3 2 x Answer: 2 log 5 3 – log 5 x.](https://reader035.fdocuments.us/reader035/viewer/2022081506/56649de85503460f94ae22bc/html5/thumbnails/34.jpg)
H.O.T. Question
• Why do we need Pascal Triangle? What is it used for?
![Page 35: Unit 2 Logarithms 10-11-12. DO NOW Expand the logarithm and simplify if possible Log 5 3 2 x Answer: 2 log 5 3 – log 5 x.](https://reader035.fdocuments.us/reader035/viewer/2022081506/56649de85503460f94ae22bc/html5/thumbnails/35.jpg)
Rewrite expressions
• Factoring by GCF• Factoring trinomials (leading coefficients)• Factoring sum & difference of cubes
![Page 36: Unit 2 Logarithms 10-11-12. DO NOW Expand the logarithm and simplify if possible Log 5 3 2 x Answer: 2 log 5 3 – log 5 x.](https://reader035.fdocuments.us/reader035/viewer/2022081506/56649de85503460f94ae22bc/html5/thumbnails/36.jpg)
examples
![Page 37: Unit 2 Logarithms 10-11-12. DO NOW Expand the logarithm and simplify if possible Log 5 3 2 x Answer: 2 log 5 3 – log 5 x.](https://reader035.fdocuments.us/reader035/viewer/2022081506/56649de85503460f94ae22bc/html5/thumbnails/37.jpg)
Terms and examples
• Zeroes, quadratics, roots, end behaviors, radicals
![Page 38: Unit 2 Logarithms 10-11-12. DO NOW Expand the logarithm and simplify if possible Log 5 3 2 x Answer: 2 log 5 3 – log 5 x.](https://reader035.fdocuments.us/reader035/viewer/2022081506/56649de85503460f94ae22bc/html5/thumbnails/38.jpg)
Rational root theorem
![Page 39: Unit 2 Logarithms 10-11-12. DO NOW Expand the logarithm and simplify if possible Log 5 3 2 x Answer: 2 log 5 3 – log 5 x.](https://reader035.fdocuments.us/reader035/viewer/2022081506/56649de85503460f94ae22bc/html5/thumbnails/39.jpg)
Higher degree functions
• Example and definitions
![Page 40: Unit 2 Logarithms 10-11-12. DO NOW Expand the logarithm and simplify if possible Log 5 3 2 x Answer: 2 log 5 3 – log 5 x.](https://reader035.fdocuments.us/reader035/viewer/2022081506/56649de85503460f94ae22bc/html5/thumbnails/40.jpg)
Wrap up