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 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:

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)}

Inverse of a function

Inverse of 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 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 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

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

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

Exponential functions Exponential growth Exponential decay

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

The natural base e

Calculator Keys Second function of divide Second function of LN (left side)

Dwight Eisenhower – American President “Pessimism never won any battle.”

Property of equivalent exponents For b>0 and b not equal to 1

Compound Interest A = Amount P = Principal r = annual interest rate in decimal form t= number of years

Continuous Compounding A = Amount P = Principal r = rate in decimal form t = number of years

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

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

Mathematics 116 – 4.2 Logarithmic Functions and Their Graphs

Definition of Logarithm

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 e

Basic Properties of logs

**Property of Logarithms One to One Property

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

Problem: change of base

Logarithm Theorems

Basic Properties of logarithms

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

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

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

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

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

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 20 th 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”

Logistic Growth Model

pH a measure of the hydrogen ion concentration of a solution.

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

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 – 2 nd degree Cubic – 3 rd degree Quartic – 4 th degree Ln (natural logarithmic logarithm) Exponential Power Logistic Sin – (trigonometric)

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

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