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

activating

• http://youtu.be/zzu2POfYv0Y?t=1s

7 – 4

• Objective: Understand the properties of logarithms

• Objective: expand and condense logarithmic expressions

• Objective: Change of base formula

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

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

One more problem

• What is the value of log 7 25? Use the change of base formula

• About 1.65

worksheets

• 7 -4 think about a plan• 7 -4 puzzle: letter scramble

7-5, 7-6, and polynomials

10-15-12

Do now

• Pg. 461 #22• Pg. 467 #70 and 72• Pg. 473 # 7

7 - 5

• How can you solve exponential equations? • Objective: solve logarithmic equations using

technology and algebraically

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

examples

• Solving an exponential equation – common base

• Pg. 469• Finding solutions• Use power property of exponents to solve

examples

• Solving an exponential equation – different bases

• Finding solutions• Solve by taking logarithm of each side of the

equation

Solving an exponential equation with graph or table

Modeling with exponential equations

• Logarithmic equation: is an equation that includes one or more logarithms involving a variable

• Pg. 477

Using logarithmic properties to solve an equation

Solving a logarithmic equation

• Problems in book and worksheet

H.O.T. question/activity/task

• Given y = ab cx

• Explain how replacing c with ( - c ) affects the function

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

Doubling time discovery

• Test thursday unit 2, complete or try to finish review packet

• http://www.regentsprep.org/Regents/math/ALGEBRA/AE7/ExpDecayL.htm

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

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.

http://www.wikihow.com/Understand-Logarithms

Begin unit 3

• Unit 3 diagnostic test• Homework if don’t get too

Unit 3 Polynomials

Agenda

1. Do now2. Activating3. What’s up next?

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?

Vocabulary

• Zeroes• Binomial expansion• Multiplicity• Relative extrema• concavity

Operations of polynomials

• Add, subtract, multiply polynomials• Synthetic division• Remainder theorem

Operations of polynomial problems

Binomial expansion

• Pascal’s triangle

Binomial

• Examples:

H.O.T. Question

• Why do we need Pascal Triangle? What is it used for?

Rewrite expressions

• Factoring by GCF• Factoring trinomials (leading coefficients)• Factoring sum & difference of cubes

examples

Terms and examples

• Zeroes, quadratics, roots, end behaviors, radicals

Rational root theorem

Higher degree functions

• Example and definitions

Wrap up