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: 1, 2 , 3, 4 5, 6 1,2 , 3,2 , 5,2 1,2 , 1,4 , 1,6 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 f a function. 1 • Designated by • and read f inverse 1 f 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 f 1,2 , 3,4 , 5,6 f 1 2,1 4,3 , 6,5 Inverse of function f 1,2 , 3,2 , 5,2 f 1 2,1 , 2,3 , 2,5 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.” x log b x log b y y xy log b x log b y 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 x 1 1 e x f ( x) b x Graphs-Determine domain, range, function, 1-1, x intercepts, y intercepts, asymptotes f ( x) 2 x Graphs-Determine domain, range, function, 1-1, x intercepts, y intercepts, asymptotes 1 g ( x) 2 x Growth and Decay f ( x) b • Growth: if b > 1 • Decay: if 0 < b < 1 x 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 x 1 as x 1 e x e 2.718281828 Calculator Keys • Second function of divide • Second function of LN (left side) e x Property of equivalent exponents • For b>0 and b not equal to 1 if b b x y then x y Compound Interest • A= amount P = Principal t = time • r = rate per year • n = number of times compounded r A P 1 n nt Compound interest problem • Find the accumulated amount in an account if $5,000 is deposited at 6% compounded quarterly for 10 years. .06 A 5000 1 4 4 10 A $9070.09 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: logb x y x b y 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 log10 x y log x y Natural logarithmic function The logarithmic function with a base of e log e x y ln x y 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 log a x logb x log a b Problem: change of base log 3 5 log10 5 log5 log10 3 log3 log e 5 ln 5 1.46 log e 3 ln 3 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 log b 1 0 log b b 1 log b x log b y x y For x>0, y>0, b>0 and b not 1 Product rule of Logarithms logb xy logb x logb y For x>0, y>0, b>0 and b not 1 Quotient rule for Logarithms x log b log b x log b y y For x>0, y>0, b>0 and b not 1 Power rule for Logarithms logb x r logb x r Objectives: • Apply the product, quotient, and power properties of logarithms. • Combine and Expand logarithmic expressions Theorems summary Logarithms: I .logb xy logb x logb y x II .log b log b x log b y y III .logb x r logb x r 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 2 x1 25 15 x 0.0794 Objective: • Solve equations containing logarithms. Sample Problem Logarithmic equation log3 2 x 5 2 x2 Sample Problem Logarithmic equation log 2 5 x 1 log 2 x 1 3 x3 Sample Problem Logarithmic equation log 2 x 2 log 2 x 3 x 4 or x 2 2 Sample Problem Logarithmic equation log5 x log5 x 3 log5 4 1 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…”