Chapter 5: 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

5.2 Exponential Functions Additional Properties of Exponents For any real number a > 0, a  0, the following statements are true: (a) ax is a unique real number for each real number x. (b) ab = ac if and only if b = c. (c) If a > 1 and m < n, then am < an. If 0 < a < 1 and m < n, then am > an.

5.2 Exponential Functions If a > 0, a  1, then f (x) = ax defines the exponential function with base a.

5.2 Graphs of Exponential Functions Example Graph 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,).

5.2 Comparing Graphs Example Explain why the graph of is a reflection across the y-axis of the graph of Analytic Solution Show that g(x) = f (–x).

5.2 Comparing Graphs Graphical Solution The graph below indicates that g(x) is a reflection across the y-axis of f (x).

5.2 Graph of f (x) = ax, a > 1

5.2 Graph of f (x) = ax, 0 < a < 1

5.2 Using Translations to Graph an Exponential Function Example Explain how the graph of is obtained from the graph of Solution

5.2 Example using Graphs to Evaluate Exponential Expressions Example Use a graph to evaluate Solution With we find that y  2.6651441 from the graph of y = 0.5x.

5.2 Exponential Equations (Type I) Example Solve Solution Write with the same base. Set exponents equal and solve.

5.2 Using a Graph to Solve Exponential Inequalities Example Solve the inequality Solution Using the graph below, the graph lies above the x-axis for values of x less than 0.5. The solution set for y > 0 is (–, 0.5).

5.2 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

5.2 Compound Interest Formula Example Suppose that $1000 is invested at an annual rate of 4%, compounded quarterly. Find the total amount in the account after 10 years if no withdrawals are made. Solution The final balance is $1488.86. Suppose that a principal of P dollars is invested at an annual interest rate r (in decimal form), compounded n times per year. Then, the amount A accumulated after t years is given by the formula

5.2 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.

5.2 Continuous Compounding Formula Example Suppose $5000 is deposited in an account paying 3% compounded continuously for 5 years. Find the total amount on deposit at the end of 5 years. Solution The final balance is $5809.17. If an amount of P dollars is deposited at a rate of interest r (in decimal form) compounded continuously for t years, then the final amount in dollars is

5.2 Modeling the Risk of Alzheimer’s Disease Example The 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. Compare the risk at age 75 with the risk at age 55. Solution x = 75 – 55 = 20, so Thus, the risk is almost fives times as great at age 75 as at age 55.