Advanced Algebra Chapter 8

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Advanced Algebra Chapter 8

• Exponential and Logarithmic Functions

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Exponential Growth8.1
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Exponential Functions

• Variable is now an exponent
• Base must be a positive Number
• Why positive?
• Base also cannot be 1
• Why?

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Exponential Functions--Graphing

• Plot Points!

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Exponential Graphing

• End Behavior
• As x goes to infinity
• As x goes to negativeinfinity

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Exponential Graphing

• Asymptote
• A line which a graphapproaches as you moveaway
  from the origin butnever actually reach

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Exponential Graphs

• The graph passes through the point ( 0 , a )
• The y-intercept is a
• The x-axis is the asymptote of the graph
• Domain All real numbers
• Range
• Greater than zero if a is positive
• Less than zero if a is negative

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Exponential Graphs

• If b is greater than 1
• Exponential Growth Function
• If b is between 0 and 1
• Exponential Decay Function

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Graphing
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Graphing
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Graphing
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Exponential Growth Models

• Basic Growth Model
• Populations
• Cost increase, etc.
• Compound Interest
• Loans
• Investments

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Compound Interest

• You purchase a brand new car for 7500. You get
  a loan for 5 years at a rate of 5.6

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p.46914,16,25-27,59-61
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Exponential Decay8.2
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Exponential Graphs

• If b is greater than 1
• Exponential Growth Function
• If b is between 0 and 1
• Exponential Decay Function

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Graphing
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Graphing
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Graphing
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Graphing
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Exponential Decay Model
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Exponential Decay

• The new car you purchased for 7500 decays in
  value. A car loses approximately 16 of its
  value every year. How much will your new car
  be worth in 2 years? When your loan is up?

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Eulers number8.3
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What is e? Where does it come from?

• Consider the following
• Try some different values of x

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What is e? Where does it come from?
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Eulers Number

• The natural number
• Named after Leonard Euler
• Irrational
• Occurs naturally in exponential growth/decay
  models
• Why?

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Eulers Number

• Consider our Exponential Growth (interest) model
• Is there part of this equation that looks
  familiar??
• SoWhat happens as the number of times our
  interest is compounded gets larger??

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Continuously Compounded Interest
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Continuously Compounded Interest

• You invest 1000 in an account that pays 8
  annual interest compounded continuously. What is
  the balance after 1 year? 10 years? 50 years?

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Prosperities of e

• e is a number like anything else
• All properties of exponents/rules still apply the
  same!

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Examples

• Simplify

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Exponential Growth or Decay

• Consider
• If exponent and leading coefficient are the same
  sign
• Growth function
• If exponent and leading cofficient are opposite
  signs
• Decay function

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Logarithmic Functions8.4
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Log Functions

• For every operation in math, there is an opposite
  operation
• Addition subtraction
• Multiplication division
• Square, Square root

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Log Function

• Inverse (Opposite) of an exponential function

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Log Functions
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Log Functions
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Log Functions
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Common Log

• Log base 10

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Natural Log

• Log base e
• Denoted as

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Properties of Logs

• RememberLogs are inverse functions so logs and
  exponents canceljust like

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Properties of Logs

• RememberLogs are inverse functions so logs and
  exponents canceljust like
• If the bases are the same, they cancel

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Properties
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p.49016-20,24-28,48-52
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Finding Inverses and Graphing Log Functions8.4
(Day 2)
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How to find inverse functions?

• Switch the x and y variables
• Solve for y

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Examples
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Examples
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Graphs

• Log functions are inverses of exponential
  functions
• Consider the graph of an exponential

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Graphs
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Graphing Logs

• The line is the vertical asymptote
• The domain is , range is all real
  numbers
• Why cant x be equal to h?
• If Graph moves up to the
  right
• If Graph moves down to the
  right

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Graphing Logs State domain and range
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Graphing Logs State domain and range
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Graphing Logs State domain and range
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More Properties of Logs8.5
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Product Property
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Product Property
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Product Property
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Quotient Property
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Quotient Property
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Quotient Property
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Power Property
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Power Property
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Power Property
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Expanding/Condensing Logs

• Bases MUST be the same
• Similarly to combining square roots/cube roots
• Must be the same

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Expanding Logs
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Expanding Logs
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Condensing Logs
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Condensing Logs
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Change of Base Formula

• Allows us to manipulate equations
• Now, we can evaluate on our calculators!
• Commonly switch to base 10

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Change of Base Formula
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Does it matter which base we choose?

• Instead of changing to base 10, could we change
  to base e?

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p.49614-71 Every 3rd
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Solving Exponential and Log Equations!8.6
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Solving

• If two powers with the same base are equal, then
  their exponents MUST be equal
• Same is true of log functionsbases must be equal

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Examples
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Examples
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Examples
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Examples
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Examples
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p.50643-59 Odd