Precalculus – MAT 129

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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

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
• Solving Exponential and Logarithmic Equations
• Exponential and Logarithmic Models
• Nonlinear Models
3.1 – Exponential Fxns and Their Graphs
• Exponential Functions
• Graphs of Exponential Functions
• The Natural Base e
• Applications
3.1 – Exponential Functions
• The exponential function f with base a is denoted by:

f(x)=ax

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

Pg. 187 Example 4

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

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

Pg. 189 Example 6

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

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

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

Pg. 191 Examples 8 and 9.

You will be responsible for knowing the compound interest formula.

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

Pg. 203 #33.

Solve the equation for x.

log7x=log79

Solution Example 1.3.2

Pg. 203 #33.

x=9

3.2 – Graphs of Logarithmic Fxns
• See beige box on pg. 199
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
• 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

Pg. 201 Example 9.

Note both the algebraic and graphical solutions.

3.2 – Application

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

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

Pg. 207 Examples 1 & 2.

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

3.3 – Properties of Logarithms

See blue box on pg. 208.

Example 2.3.3

Pg. 208 Example 3

These should be pretty self explanatory.

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

Pg. 209 Examples 5&6.

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

3.4 – Solving Exponential and Logarithmic Equations
• Introduction
• Solving Exponential Equations
• Solving Logarithmic Equations
• Applications
3.5 –Exponential and Logarithmic Models
• Introduction
• Exponential Growth and Decay
• Gaussian Models
• Logistic Growth Models
• Logarithmic Models
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 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
• 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

In Class QUIZ:

pp. 234

#30, 41a, 42a.