Intermediate Algebra Chapter 9 Exponential and Logarithmic Functions

Intermediate Algebra Chapter 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:

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

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

Inverse of function

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

Graphs-Determine domain, range, function, 1 -1, x intercepts, y intercepts, asymptotes

Growth and Decay • Growth: if b > 1 • Decay: if 0 < b < 1

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

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

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

Compound Interest • A= amount P = Principal t = time • r = rate per year • n = number of times compounded

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

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:

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

Natural logarithmic function The logarithmic function with a base of e

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

Problem: change of base

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

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: • Apply the product, quotient, and power properties of logarithms. • Combine and Expand logarithmic expressions

Theorems summary Logarithms:

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

Objective: • Solve equations containing logarithms.

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. ”

Galileo Galilei (1564 -1642) • “The universe…is written in the language of mathematics…”