Mathematics 116Chapter 5

•Exponential

•And

•Logarithmic Functions

John Quincy Adams

• “Patience and perseverance have a magical effect before which difficulties disappear and obstacles vanish.”

• Mathematics 116

• Exponential Functions• and

• Their Graphs

Def: Relation• A relation is a set of ordered pairs.• Designated by:

• Listing• Graphs• Tables• Algebraic equation• Picture• Sentence

Def: Function

• A function is a set of ordered pairs in which no two different ordered pairs have the same first component.

• Vertical line test – used to determine whether a graph represents a function.

Defs: domain and range

• Domain: The set of first components of a relation.

• Range: The set of second components of a relation

Examples of Relations:

1,2 , 3,4 5,6

1,2 , 3,2 , 5,2

1,2 , 1,4 , 1,6

Objectives

• Determine the domain, range of relations.

• Determine if relation is a function.

Mathematics 116

•Inverse Functions

Objectives:

• Determine the inverse of a function whose ordered pairs are listed.

• Determine if a function is one to one.

Inverse Function

• g is the inverse of f if the domains and ranges are interchanged.

• f = {(1,2),(3,4), (5,6)}

• g= {(2,1), (4,3),(6,5)}

1( ) ( )g x f x

Inverse of a function

1,2 , 3,4 , 5,6f

1 2,1 4,3 , 6,5f

Inverse of function

1,2 , 3,2 , 5,2f

1 2,1 , 2,3 , 2,5f

One-to-One Function

• A function f is one-to one if for and and b in its domain, f(a) = f(b) implies a = b.

• Other – each component of the range is unique.

One-to-One function

• Def: A function is a one-to-one function if no two different ordered pairs have the same second coordinate.

Horizontal Line TestA test for one-to one

• If a horizontal line intersects the graph of the function in more than one point, the function is not one-to one

Existence of an Inverse Function

• A function f has an inverse function if and only if f is one to one.

Find an Inverse Function

• 1. Determine if f has an inverse function using horizontal line test.

• 2. Replace f(x) with y

• 3. Interchange x and y

• 4. Solve for y

• 5. Replace y with 1( )f x

Definition of Inverse Function

• Let f and g be two functions such that

f(g(x))=x for every x in the domain of g

and g(f(x))=x for every x in the domain of.

• g is the inverse function of the function f

Objective

• Recognize and evaluate exponential functions

with base b.

Michael Crichton – The Andromeda Strain

(1971)• The mathematics of uncontrolled

growth are frightening. A single cell of the bacterium E. coli would, under ideal circumstances, divide every twenty minutes. It this way it can be shown that in a single day, one cell of E. coli could produce a super-colony equal in size and weight to the entire planet Earth.”

Graph

• Determine: Domain, range, function and why, 1-1 function and why, y intercept, x intercept, asymptotes

( ) 2xf x

Graph

• Determine: Domain, range, function and why, 1-1 function and why, y intercept, x intercept, asymptotes

1( )

2

x

f x

Exponential functions

• Exponential growth

• Exponential decay

( ) , 1xf x b b

( ) ,0 1xf x b b

Properties of graphs of exponential functions

• Function and 1 to 1

• y intercept is (0,1) and no x intercept(s)

• Domain is all real numbers

• Range is {y|y>0}

• Graph approaches but does not touch x axis – x axis is asymptote

• Growth or decay determined by base

The Natural Base e

2.718281828e

The natural base e

1lim 1

n

as nn

Calculator Keys

• Second function of divide

• Second function of LN (left side) xe

Dwight Eisenhower – American President

•“Pessimism never won any battle.”

Property of equivalent exponents

• For b>0 and b not equal to 1

x yif b b

then x y

Compound Interest

• A = Amount

• P = Principal

• r = annual interest rate in decimal form

• t= number of years

1nt

rA P

n

Continuous Compounding

• A = Amount

• P = Principal

• r = rate in decimal form

• t = number of years

rtA Pe

Compound interest problem

• Find the accumulated amount in an account if $5,000 is deposited at 6% compounded quarterly for 10 years.

4 10.06

5000 14

A

$9070.09A

Objectives

• Recognize and evaluate exponential functions with base b

• Graph exponential functions

• Recognize, evaluate, and graph exponential functions with base e.

• Use exponential functions to model and solve real-life problems.

Albert Einstein – early 20th century physicist

• “Everything should be made as simple as possible, but not simpler.”

Mathematics 116 – 4.2

• Logarithmic Functions

• and

• Their Graphs

Definition of Logarithm

log yb x y x b

Objectives• Recognize and evaluate

logarithmic function with base b• Note: this includes base 10 and

base e• Graph logarithmic functions

–By Hand–By Calculator

Shape of logarithmic graphs

• For b > 1, the graph rises from left to right.

• For 0 < b < 1, the graphs falls from left to right.

Properties of Logarithmic Function

• Domain:{x|x>0}• Range: all real numbers• x intercept: (1,0)• No y intercept• Approaches y axis as vertical

asymptote• Base determines shape.

Evaluate Logs on calculator

• Common Logs – base of 10

• Natural logs – base of e10log logx x

log lne x x

Basic Properties of logs

0

1

log 1 0 1

log 1

log

b

b

xb

b

b b b

b x

**Property of Logarithms

• One to One Property

log logb bIf x y

then x y

Objective

• Use logarithmic functions to model and solve real-life problems.

Jim Rohn

• “You must take personal responsibility. You cannot change the circumstances, the seasons, or the wind, but you can change yourself. That is something you have charge of.”

Mathematics 116 – 4.3

•Properties

•of

•Logarithms

Change of Base Formula

loglog

logb

ab

xx

a

Problem: change of base

3log 5 10

10

log 5 log5

log 3 log3

log 5 ln5

log 3 ln3e

e

1.46

Logarithm Theorems

log log log

log log log

log log

b b b

b b b

rb b

xy x y

xx y

y

x r x

Basic Properties of logarithms

log 1 0b log 1b b

log logb bx y x y

For x>0, y>0, b>0 and b not 1Product rule of Logarithms

log log logb b bxy x y

For x>0, y>0, b>0 and b not 1Quotient rule for Logarithms

log log logb b b

xx y

y

For x>0, y>0, b>0 and b not 1Power rule for Logarithms

log logrb bx r x

Objectives:

• Use properties of logarithms to evaluate or rewrite logarithmic expressions

• Use properties of logarithms to expand logarithmic expressions

• Use properties of logarithms to condense logarithmic expressions.

Albert Einstein

• “The important thing is not to stop questioning.”

Mathematics 116

• Solving

• Exponential

• and

• Logarithmic Equations

Solving Exponential Equations

• 1. *** Rewrite equation so exponential term is isolated.

• 2. Rewrite in logarithmic form

• Use base ln if base is e.

• 3. Solve the equation

• 4. Check the results– Graphically or algebraically

Exponential equation

2 125 15x 0.0794x

Solve Logarithmic Equations

• 1. *** Rewrite equation so logarithmic term is isolated. Or use one-one property

• 2. Rewrite in exponential form

• 3. Solve the equation

• 4. Check the results– Graphically or algebraically

Sample Problem Logarithmic equation

3log 2 5 2x

2x

Sample Problem Logarithmic equation

2 2log 5 1 log 1 3x x

3x

Sample Problem Logarithmic equation

2 2log 2 log 3x x

4 2x or x 2

Sample Problem Logarithmic equation

5 5 5log log 3 log 4x x

1

Walt Disney

• “Disneyland will never be completed. It will continue to grow as long as there is imagination left in the world.”

Objectives:

• Solve exponential equations

• Solve logarithmic equations

• Use exponential and logarithmic equations to model and solve real-life problems.

Hans Hofmann – early 20th century teacher and painter

• “The ability to simplify means to eliminate the unnecessary so that the necessary may speak.”

Mathematics 116

• Exponential

• and

• Logarithmic

• Models

Objective

• Recognize the most common types of models involving exponential or logarithmic functions

Models

• Exponential growth• Exponential decay• Logarithmic

–Common logs–Natural logs

Gaussian Model

• “normal curve”

2 /x b cy ae

Logistic Growth Model

1 rx

ay

be

pHa measure of the hydrogen ion

concentration of a solution.

10logpH H

Magnitude of Earthquake

• Uses Richter scale I is intensity which is a measure of the wave energy of an earthquake

100

0

log

1

IR

I

I

Carl Zuckmeyer

• “One-half of life is luck; the other half is discipline – and that’s the important half, for without discipline you wouldn’t know what t do with luck.”

Mathematics 116 – 4.6

• Exploring Data:

• Nonlinear Models

Objectives

• Classify Scatter Plots

• Use scatter plots and a graphing calculator to find models for data and choose a model that best fits a set of data.

• Use a graphing utility to find models to fit data.

• Make predictions from models.

Calculator regression models

• Linear(mx+b) (preferred) and (b+mx)• Quadratic – 2nd degree• Cubic – 3rd degree• Quartic – 4th degree• Ln (natural logarithmic logarithm)• Exponential• Power• Logistic• Sin – (trigonometric)

Julie Andrews

•“Perseverance is failing 19 times and succeeding the 20th.”

Walt Disney

• “Disneyland will never be completed. It will continue to grow as long as there is imagination left in the world.”