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Intermediate Algebra Chapter 9 Exponential and Logarithmic Functions.

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Presentation on theme: "Intermediate Algebra Chapter 9 Exponential and Logarithmic Functions."— Presentation transcript:

1 Intermediate Algebra Chapter 9 Exponential and Logarithmic Functions

2 Intermediate Algebra Review of Functions

3 Def: Relation A relation is a set of ordered pairs. Designated by: Listing Graphs Tables Algebraic equation Picture Sentence

4 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.

5 Defs: domain and range Domain: The set of first components of a relation. Range: The set of second components of a relation

6 Examples of Relations:

7 Objectives Determine the domain, range of relations. Determine if relation is a function.

8 Intermediate Algebra 9.2 Inverse Functions

9 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

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

11 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.

12 Inverse of a function

13 Inverse of function

14 Objectives: Determine the inverse of a function whose ordered pairs are listed. Determine if a function is one to one.

15 Intermediate Algebra 9.3 Exponential Functions

16 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.”

17 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

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

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

21 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

22 Natural Base e

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

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

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

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

27 Objectives: Determine and graph exponential functions. Use the natural base e Use the compound interest formula.

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

29 Intermediate Algebra 9.4,9.5,9.6 Logarithmic Functions

30 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:

31 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.

32 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.

33 Common Logarithmic Function The logarithmic function with base 10

34 Natural logarithmic function The logarithmic function with a base of e

35 Calculator Keys [LOG] [LN]

36 Objective: Determine the common log or natural log of any number in the domain of the logarithmic function.

37 Change of Base Formula For x > 0 for any positive bases a and b

38 Problem: change of base

39 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.

40 Intermediate Algebra 10.5 Properties of Logarithms

41 Basic Properties of logarithms

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

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

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

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

46 Theorems summary Logarithms:

47 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.”

48 Intermediate Algebra 9.7 Exponential and Logarithmic Equations

49 Objective: Solve equations that have variables as exponents.

50 Exponential equation

51 Objective: Solve equations containing logarithms.

52 Sample Problem Logarithmic equation

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

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58 Galileo Galilei ( ) “The universe…is written in the language of mathematics…”


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