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Precalculus – MAT 129. Instructor: Rachel Graham Location: BETTS Rm. 107 Time: 8 – 11:20 a.m. MWF. Chapter Three. Exponential and Logarithmic Functions. Ch. 3 Overview. Exponential Fxns and Their Graphs Logarithmic Fxns and Their Graphs Properties of Logarithms

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precalculus mat 129

Precalculus – MAT 129

Instructor: Rachel Graham

Location: BETTS Rm. 107

Time: 8 – 11:20 a.m. MWF

chapter three

Chapter Three

Exponential and Logarithmic Functions

ch 3 overview
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
3 1 exponential fxns and their graphs
3.1 – Exponential Fxns and Their Graphs
  • Exponential Functions
  • Graphs of Exponential Functions
  • The Natural Base e
  • Applications
3 1 exponential functions
3.1 – Exponential Functions
  • The exponential function f with base a is denoted by:

f(x)=ax

3 1 graphs of exponential fxns
3.1 – Graphs of Exponential Fxns
  • Figure 3.1 on pg. 185 shows the form of the graph of:

y=ax

  • Figure 3.2 on pg. 185 shows the form of the graph of:

y=a-x

example 1 3 1
Example 1.3.1

Pg. 187 Example 4

After looking at the solution read the paragraph at the bottom of the page.

3 1 the natural base e
3.1 – The Natural Base e
  • e≈2.71828
    • Useful for a base in many situations.
  • f(x)=ex is called the natural exponential function.
example 2 3 1
Example 2.3.1

Pg. 189 Example 6

Be sure you know how to evaluate this function on your calculator.

3 1 applications
3.1 – Applications
  • The most widely used application of the exponential function is for showing investment earnings with continuously compounded interest.
formulas for compounding interest
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: A=P(1+r/n)nt
  • For continuous compounding: A=Pert
example 3 3 1
Example 3.3.1

Pg. 191 Examples 8 and 9.

You will be responsible for knowing the compound interest formula.

activities 191
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.

3 2 logarithmic fxns and their graphs
3.2 – Logarithmic Fxns and Their Graphs
  • Logarithmic Functions
  • Graphs of Logarithmic Functions
  • The Natural Logarithmic Function
  • Applications
3 2 logarithmic functions
3.2 – Logarithmic Functions
  • The inverse of the exponential function is the logarithmic function.

For x>0, a>0, and a≠1,

y=logax if and only if x=ay.

f(x)=logax is called the logarithmic function with base a.

properties of logarithms
Properties of Logarithms
  • loga1=0 because a0=1.
  • logaa=1 because a1=a.
  • logaax =x because alogx=x.
  • If logax=logay, then x=y
example 1 3 2
Example 1.3.2

Pg. 203 #33.

Solve the equation for x.

log7x=log79

solution example 1 3 2
Solution Example 1.3.2

Pg. 203 #33.

x=9

3 2 graphs of logarithmic fxns
3.2 – Graphs of Logarithmic Fxns
  • See beige box on pg. 199
3 2 the natural logarithmic fxn
3.2 – The Natural Logarithmic Fxn

For x>0,

y=ln x if and only if x=ey.

f(x) = logex = ln x is called the natural logarithmic function.

properties of natural logarithms
Properties of Natural Logarithms
  • ln 1=0 because e0=1.
  • ln e=1 because e1=e.
  • ln ex =x because elnx=x.
  • If ln x=ln y, then x=y
example 2 3 2
Example 2.3.2

Pg. 201 Example 9.

Note both the algebraic and graphical solutions.

3 2 application
3.2 – Application

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

3 3 properties of logarithms
3.3 – Properties of Logarithms
  • Change of Base
  • Properties of Logarithms
  • Rewriting Logarithmic Expressions
3 3 change of base
3.3 – Change of Base

To evaluate logarithms at different bases you can use the change of base formula:

logax = (logbx/ logba)

example 1 3 3
Example 1.3.3

Pg. 207 Examples 1 & 2.

Note both log and ln functions will yield the same result.

3 3 properties of logarithms27
3.3 – Properties of Logarithms

See blue box on pg. 208.

example 2 3 3
Example 2.3.3

Pg. 208 Example 3

These should be pretty self explanatory.

3 3 rewriting log fxns
3.3 – Rewriting Log Fxns
  • This is where you use the multiplication, division, and power rules to expand and condense logarithmic expressions.
example 3 3 3
Example 3.3.3

Pg. 209 Examples 5&6.

Note that a square root is equal to the power of ½.

3 4 solving exponential and logarithmic equations
3.4 – Solving Exponential and Logarithmic Equations
  • Introduction
  • Solving Exponential Equations
  • Solving Logarithmic Equations
  • Applications
3 5 exponential and logarithmic models
3.5 –Exponential and Logarithmic Models
  • Introduction
  • Exponential Growth and Decay
  • Gaussian Models
  • Logistic Growth Models
  • Logarithmic Models
example 1 3 5
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?

example 2 3 5
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?
example 3 3 5
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.

activities
Activities

In Class QUIZ:

pp. 234

#30, 41a, 42a.