intermediate algebra chapter 9 exponential and logarithmic functions
TRANSCRIPT
Intermediate AlgebraChapter 9
•Exponential
•and
•Logarithmic Functions
Intermediate Algebra 9.1-9.2
• Review of Functions
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.
Intermediate Algebra 9.2
•Inverse Functions
Inverse of a function
• The inverse of a function is determined by interchanging the domain and the range of the original function.
• The inverse of a function is not necessarily a function.
• Designated by
• and read f inverse
1f
1f
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 Test
• A function is a one-to-one function if and only if no horizontal line intersects the graph of the function at more than one point.
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
Objectives:
• Determine the inverse of a function whose ordered pairs are listed.
• Determine if a function is one to one.
Intermediate Algebra 9.3
•Exponential Functions
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.”
Definition of Exponential Function
• If b>0 and b not equal to 1 and x is any real number, an exponential function is written as
2 2log 2 log 3
1
2
0.0794
.log log log
.log log log
11
b b b
b b b
x
x x
x
xII x y
y
I xy x y
as x ex
( ) xf x b
Graphs-Determine domain, range, function, 1-1, x intercepts,
y intercepts, asymptotes
( ) 2xf x
Graphs-Determine domain, range, function, 1-1, x intercepts,
y intercepts, asymptotes
1( )
2
x
g x
Growth and Decay
•Growth: if b > 1
•Decay: if 0 < b < 1
( ) xf x 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
Natural Base e
11
x
as x ex
2.718281828e
Calculator Keys
• Second function of divide
• Second function of LN (left side) xe
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 t = time
• r = rate per year
• n = number of times compounded
1nt
rA P
n
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:
• Determine and graph exponential functions.
• Use the natural base e• Use the compound interest
formula.
Dwight Eisenhower – American President
•“Pessimism never won any battle.”
Intermediate Algebra 9.4,9.5,9.6
•Logarithmic Functions
Definition: Logarithmic Function
• For x > 0, b > 0 and b not equal to 1 toe logarithm of x with base b is defined by the following:
log yb x y x b
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.
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.
Common Logarithmic Function The logarithmic function with
base 10
10log logx y x y
Natural logarithmic functionThe logarithmic function with a
base of e
log lne x y x y
Calculator Keys
• [LOG]
• [LN]
Objective:
• Determine the common log or natural log of any number in the domain of the logarithmic function.
Change of Base Formula
• For x > 0 for any positive bases a and b
loglog
loga
ba
xx
b
Problem: change of base
3log 5 10
10
log 5 log5
log 3 log3
log 5 ln5
log 3 ln3e
e
1.46
Objective
• Use the change of base formula to determine an approximation to the logarithm of a number when the base is not 10 or e.
Intermediate Algebra 10.5
•Properties
•of
•Logarithms
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:
• Apply the product, quotient, and power properties of logarithms.
• Combine and Expand logarithmic expressions
Theorems summary Logarithms:
.log log logb b bI xy x y
.log log logb b b
xII x y
y
.log logrb bIII x r x
Norman Vincent Peale
• “Believe it is possible to solve your problem. Tremendous things happen to the believer. So believe the answer will come. It will.”
Intermediate Algebra 9.7
•Exponential •and
•Logarithmic •Equations
Objective:
• Solve equations that have variables as exponents.
Exponential equation
2 125 15x 0.0794x
Objective:
•Solve equations containing logarithms.
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.”
Galileo Galilei (1564-1642)
•“The universe…is written in the language of mathematics…”