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Mathematics 116 Chapter 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.

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mathematics 116 chapter 5
Mathematics 116Chapter 5
  • Exponential
  • And
  • Logarithmic Functions
john quincy adams
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
Def: Relation
  • A relation is a set of ordered pairs.
  • Designated by:
  • Listing
  • Graphs
  • Tables
  • Algebraic equation
  • Picture
  • Sentence
def function
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
Defs: domain and range
  • Domain: The set of first components of a relation.
  • Range: The set of second components of a relation
objectives
Objectives
  • Determine the domain, range of relations.
  • Determine if relation is a function.
mathematics 116
Mathematics 116
  • Inverse Functions
objectives1
Objectives:
  • Determine the inverse of a function whose ordered pairs are listed.
  • Determine if a function is one to one.
inverse function
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)}
one to one function
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 function1
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 test for one to one
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
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
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
definition of inverse function
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
Objective
  • Recognize and evaluate exponential functions with base b.
michael crichton the andromeda strain 1971
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
Graph
  • Determine: Domain, range, function and why, 1-1 function and why, y intercept, x intercept, asymptotes
graph1
Graph
  • Determine: Domain, range, function and why, 1-1 function and why, y intercept, x intercept, asymptotes
exponential functions
Exponential functions
  • Exponential growth
  • Exponential decay
properties of graphs of exponential functions
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
calculator keys
Calculator Keys
  • Second function of divide
  • Second function of LN (left side)
dwight eisenhower american president
Dwight Eisenhower – American President
  • “Pessimism never won any battle.”
property of equivalent exponents
Property of equivalent exponents
  • For b>0 and b not equal to 1
compound interest
Compound Interest
  • A = Amount
  • P = Principal
  • r = annual interest rate in decimal form
  • t= number of years
continuous compounding
Continuous Compounding
  • A = Amount
  • P = Principal
  • r = rate in decimal form
  • t = number of years
compound interest problem
Compound interest problem
  • Find the accumulated amount in an account if $5,000 is deposited at 6% compounded quarterly for 10 years.
objectives2
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 20 th century physicist
Albert Einstein – early 20th century physicist
  • “Everything should be made as simple as possible, but not simpler.”
mathematics 116 4 2
Mathematics 116 – 4.2
  • Logarithmic Functions
  • and
  • Their Graphs
objectives3
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
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
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
Evaluate Logs on calculator
  • Common Logs – base of 10
  • Natural logs – base of e
property of logarithms
**Property of Logarithms
  • One to One Property
objective1
Objective
  • Use logarithmic functions to model and solve real-life problems.
jim rohn
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
Mathematics 116 – 4.3
  • Properties
  • of
  • Logarithms
objectives4
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
Albert Einstein
  • “The important thing is not to stop questioning.”
mathematics 1161
Mathematics 116
  • Solving
  • Exponential
  • and
  • Logarithmic Equations
solving exponential 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
solve logarithmic equations
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
walt disney
Walt Disney
  • “Disneyland will never be completed. It will continue to grow as long as there is imagination left in the world.”
objectives5
Objectives:
  • Solve exponential equations
  • Solve logarithmic equations
  • Use exponential and logarithmic equations to model and solve real-life problems.
hans hofmann early 20 th century teacher and painter
Hans Hofmann – early 20th century teacher and painter
  • “The ability to simplify means to eliminate the unnecessary so that the necessary may speak.”
mathematics 1162
Mathematics 116
  • Exponential
  • and
  • Logarithmic
  • Models
objective2
Objective
  • Recognize the most common types of models involving exponential or logarithmic functions
models
Models
  • Exponential growth
  • Exponential decay
  • Logarithmic
    • Common logs
    • Natural logs
gaussian model
Gaussian Model
  • “normal curve”
magnitude of earthquake
Magnitude of Earthquake
  • Uses Richter scale I is intensity which is a measure of the wave energy of an earthquake
carl zuckmeyer
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
Mathematics 116 – 4.6
  • Exploring Data:
  • Nonlinear Models
objectives6
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
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
Julie Andrews
  • “Perseverance is failing 19 times and succeeding the 20th.”
walt disney1
Walt Disney
  • “Disneyland will never be completed. It will continue to grow as long as there is imagination left in the world.”