Precalculus MAT 129

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Precalculus MAT 129

• Instructor Rachel Graham
• Location BETTS Rm. 107
• Time 8 1120 a.m. MWF

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Chapter Three

• Exponential and Logarithmic Functions

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Ch. 3 Overview

• Exponential Fxns and Their Graphs
• Logarithmic Fxns and Their Graphs
• Properties of Logarithms
• Solving Exponential and Logarithmic Equations
• Exponential and Logarithmic Models
• Nonlinear Models

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3.1 Exponential Fxns and Their Graphs

• Exponential Functions
• Graphs of Exponential Functions
• The Natural Base e
• Applications

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3.1 Exponential Functions

• The exponential function f with base a is denoted
  by
• f(x)ax

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3.1 Graphs of Exponential Fxns

• Figure 3.1 on pg. 185 shows the form of the graph
  of
• yax
• Figure 3.2 on pg. 185 shows the form of the graph
  of
• ya-x

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Example 1.3.1

• Pg. 187 Example 4
• After looking at the solution read the paragraph
  at the bottom of the page.

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3.1 The Natural Base e

• e2.71828
• Useful for a base in many situations.
• f(x)ex is called the natural exponential
  function.

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Example 2.3.1

• Pg. 189 Example 6
• Be sure you know how to evaluate this function
  on your calculator.

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3.1 Applications

• The most widely used application of the
  exponential function is for showing investment
  earnings with continuously compounded interest.

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Formulas for Compounding Interest

• After t years, the balance A in an account with
  principal P and annual interest rate r (in
  decimal form) is given by the following formulas
• For n compoundings per year AP(1r/n)nt
• For continuous compounding APert

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Example 3.3.1

• Pg. 191 Examples 8 and 9.
• You will be responsible for knowing the compound
  interest formula.

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Activities (191)

• 1. Determine the balance A at the end of 20
  years if 1500 is invested at 6.5 interest and
  the interest is compounded (a) quarterly and (b)
  continuously.
• 2. Determine the amount of money that should be
  invested at 9 interest, compounded monthly, to
  produce a final balance of 30,000 in 15 years.

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3.2 Logarithmic Fxns and Their Graphs

• Logarithmic Functions
• Graphs of Logarithmic Functions
• The Natural Logarithmic Function
• Applications

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3.2 Logarithmic Functions

• The inverse of the exponential function is the
  logarithmic function.
• For xgt0, agt0, and a?1,
• ylogax if and only if xay.
• f(x)logax is called the logarithmic function
  with base a.

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Properties of Logarithms

• loga10 because a01.
• logaa1 because a1a.
• logaax x because alogxx.
• If logaxlogay, then xy

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Example 1.3.2

• Pg. 203 33.
• Solve the equation for x.
• log7xlog79

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Solution Example 1.3.2

• Pg. 203 33.
• x9

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3.2 Graphs of Logarithmic Fxns

• See beige box on pg. 199

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3.2 The Natural Logarithmic Fxn

• For xgt0,
• yln x if and only if xey.
• f(x) logex ln x is called the natural
  logarithmic function.

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Properties of Natural Logarithms

• ln 10 because e01.
• ln e1 because e1e.
• ln ex x because eln xx.
• If ln xln y, then xy

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Example 2.3.2

• Pg. 201 Example 9.
• Note both the algebraic and graphical solutions.

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3.2 Application

• See example 10 on pg. 202 for the best
  application of logarithmic functions.

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3.3 Properties of Logarithms

• Change of Base
• Properties of Logarithms
• Rewriting Logarithmic Expressions

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3.3 Change of Base

• To evaluate logarithms at different bases you can
  use the change of base formula
• logax (logbx/ logba)

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Example 1.3.3

• Pg. 207 Examples 1 2.
• Note both log and ln functions will yield the
  same result.

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3.3 Properties of Logarithms

• See blue box on pg. 208.

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Example 2.3.3

• Pg. 208 Example 3
• These should be pretty self explanatory.

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3.3 Rewriting Log Fxns

• This is where you use the multiplication,
  division, and power rules to expand and condense
  logarithmic expressions.

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Example 3.3.3

• Pg. 209 Examples 56.
• Note that a square root is equal to the power of
  ½.

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3.4 Solving Exponential and Logarithmic
Equations

• Introduction
• Solving Exponential Equations
• Solving Logarithmic Equations
• Applications

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3.5 Exponential and Logarithmic Models

• Introduction
• Exponential Growth and Decay
• Gaussian Models
• Logistic Growth Models
• Logarithmic Models

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The Models
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Example 1.3.5

• Example 2 on pg. 227
• In a research experiment, a population of fruit
  flies is increasing according to the law of
  exponential growth. After 2 days there are 100
  fruit flies, and after 4 days there are 300 fruit
  flies. How many flies will there be after 5 days?

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Example 2.3.5

• Example 5 on pg. 230
• On a college campus of 5000 students, one student
  returns from vacation with a contagious flu
  virus. The spread of the virus is modeled on pg.
  230 where y is the total number infected after t
  days. The college will cancel classes when 40 or
  more are infected.
• How many students are infected after 5 days?
• After how many days will the college cancel
  classes?

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Example 3.3.5

• On the Richter scale, the magnitude R of an
  earthquake of intensity I is given by
• R log10 I/I0
• where I0 1 is the minimum intensity used for
  comparison. Intensity is a measure of wave energy
  of an earthquake.

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Activities

• In Class QUIZ
• pp. 234
• 30, 41a, 42a.