Copyright © 2007 Pearson Education, Inc. Slide 5-1
Copyright © 2007 Pearson Education, Inc. Slide 5-2 Chapter 5: Inverse, Exponential and Logarithmic Functions 5.1 Inverse Functions 5.2 Exponential Functions 5.3 Logarithms and Their Properties 5.4 Logarithmic Functions 5.5 Exponential and Logarithmic Equations and Inequalities 5.6 Further Applications and Modeling with Exponential and Logarithmic Functions
Copyright © 2007 Pearson Education, Inc. Slide Exponential Functions Additional Properties of Exponents –a x is a unique real number. –a b = a c if and only if b = c. –If a > 1 and m < n, then a m < a n. –If 0 a n. If a > 0, a 1, then f (x) = a x is the exponential function with base a.
Copyright © 2007 Pearson Education, Inc. Slide Graphs of Exponential Functions ExampleGraph Determine the domain and range of f. Solution There is no x-intercept. Any number to the zero power is 1, so the y-intercept is (0,1). The domain is (– , ), and the range is (0, ).
Copyright © 2007 Pearson Education, Inc. Slide Graph of f (x) = a x, a > 1
Copyright © 2007 Pearson Education, Inc. Slide Graph of f (x) = a x, 0 < a < 1
Copyright © 2007 Pearson Education, Inc. Slide Comparing Graphs ExampleExplain why the graph of is a reflection across the y-axis of the graph of Analytic Solution Show that g(x) = f (–x).
Copyright © 2007 Pearson Education, Inc. Slide Comparing Graphs Graphical Solution The graph below indicates that g(x) is a reflection across the y-axis of f (x).
Copyright © 2007 Pearson Education, Inc. Slide Using Translations to Graph an Exponential Function ExampleExplain how the graph of is obtained from the graph of Solution
Copyright © 2007 Pearson Education, Inc. Slide Example using Graphs to Evaluate Exponential Expressions Example Use a graph to evaluate SolutionWith we find that y from the graph of y =.5 x.
Copyright © 2007 Pearson Education, Inc. Slide Exponential Equations (Type I) ExampleSolve Solution Write with the same base. Set exponents equal and solve.
Copyright © 2007 Pearson Education, Inc. Slide Using a Graph to Solve Exponential Inequalities ExampleSolve the inequality SolutionUsing the graph below, the graph lies above the x-axis for values of x less than.5. The solution set for y > 0 is (– ,.5).
Copyright © 2007 Pearson Education, Inc. Slide The Natural Number e Named after Swiss mathematician Leonhard Euler e involves the expression e is an irrational number Since e is an important base, calculators are programmed to find powers of e.
Copyright © 2007 Pearson Education, Inc. Slide Compound Interest Recall simple earned interest where –P is the principal (or initial investment), –r is the annual interest rate, and –t is the number of years. If A is the final balance at the end of each year, then
Copyright © 2007 Pearson Education, Inc. Slide Compound Interest Formula ExampleSuppose that $1000 is invested at an annual rate of 8%, compounded quarterly. Find the total amount in the account after 10 years if no withdrawals are made. Solution The final balance is $ Suppose that a principal of P dollars is invested at an annual interest rate r (in percent), compounded n times per year. Then, the amount A accumulated after t years is given by the formula
Copyright © 2007 Pearson Education, Inc. Slide Continuous Compounding Formula Example Suppose $5000 is deposited in an account paying 8% compounded continuously for 5 years. Find the total amount on deposit at the end of 5 years. Solution The final balance is $ If P dollars is deposited at a rate of interest r compounded continuously for t years, the final amount A in dollars on deposit is
Copyright © 2007 Pearson Education, Inc. Slide Modeling the Risk of Alzheimer’s Disease ExampleThe chances of dying of influenza or pneumonia increase exponentially after age 55 according to the function defined by where r is the risk (in decimal form) at age 55 and x is the number of years greater than 55. What is the risk at age 75? Solutionx = 75 – 55 = 20, so Thus, the risk is almost fives times as great at age 75 as at age 55.