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Intermediate Algebra Chapter 9 Exponential and Logarithmic Functions

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Intermediate Algebra 9.1-9.2 Review of Functions

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

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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.

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

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Examples of Relations:

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

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Intermediate Algebra 9.2 Inverse Functions

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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

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

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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.

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Inverse of a function

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Inverse of function

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

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Intermediate Algebra 9.3 Exponential Functions

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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.”

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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

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

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

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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

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Natural Base e

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

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

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

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

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

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

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Intermediate Algebra 9.4,9.5,9.6 Logarithmic Functions

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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:

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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.

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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.

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Common Logarithmic Function The logarithmic function with base 10

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Natural logarithmic function The logarithmic function with a base of e

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Calculator Keys [LOG] [LN]

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

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

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Problem: change of base

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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.

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Intermediate Algebra 10.5 Properties of Logarithms

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Basic Properties of logarithms

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

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

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

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

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Theorems summary Logarithms:

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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.”

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Intermediate Algebra 9.7 Exponential and Logarithmic Equations

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Objective: Solve equations that have variables as exponents.

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Exponential equation

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Objective: Solve equations containing logarithms.

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Sample Problem Logarithmic equation

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

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