Review of Functions A relation is a function if each input has exactly output. The graph of a function passes the vertical line test.

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1 CA-Fall 011-Jordan College Algebra, 4 th edition, Beecher/Penna/Bittinger, Pearson/Addison Wesley, 01 Chapter 5: Exponential Functions and Logarithmic Functions 1 Section 5.1 Inverse Functions Inverse Relation Interchange the variables to produce an inverse relation. Graphs of a relation and its inverse are always reflections of each other across the line y = x. Example 1 Find the inverse of the relation. {(0, 1), (5, 6), (-, -4)} Review of Functions A relation is a function if each input has exactly output. The graph of a function passes the vertical line test. One-to-One Functions A function f is one-to-one if each output only maps back to one input One-to-one function. Not a one-to-one function. Horizontal Line Test If it is possible for a horizontal line to intersect the graph of a function more than once, then the function is not one-to-one and its inverse is not a. If a function is one-to-one, then it has an inverse function. Example Determine whether or not the following functions are one-to-one. a) f(x) = 4 x b) f ( x) x c) f ( x) x 3

2 Inverses of Functions If f(x) is one-to-one, then it has an inverse function named f inverse of x and the inverse function is denoted by f -1 (x). The negative 1 in f -1 is not an exponent, but is instead a name. f -1 is read f-inverse. Finding a Formula for an Inverse Function Replace f(x) with y. Interchange x and y. Solve for y. Replace y with f -1 (x). Example 3 f(x) = 3x + 7 a) Determine whether the function is one-to-one. b) If the function is one-to-one, find a formula for its inverse function. Example 4 f(x) = x 5 a) Determine whether the function is one-to-one. b) If the function is one-to-one, find a formula for its inverse function. Example 5 f(x) = -x 3 a) Determine whether the function is one-to-one. b) If the function is one-to-one, find a formula for its inverse function. Properties of One-to-One Functions and Inverses The domain of f is the range of f -1. The range of f is the domain of f -1.

3 Graphing a Function and its Inverse The graphs of a function and its inverse are reflections of each other across the line. 3 If (a, b) is a point on the graph of f(x), then the point (b, a) will be a point on the graph of f -1 (x). 3 Example 6 f ( x) x 1 a) Find the inverse of f(x). b) Graph both f and f -1 on the same set of axes. c) Give the domain and range of f and f -1. Inverse Functions and Composition for each x in the domain of f, and for each x in the domain of f -1 Example 7 Given that f(x) = 7x and f -1 (x) = (x + )/7, find.

4 Section 5. Exponential Functions and Graphs 4 Exponential Function f(x) = a x, where x is a real number, a > 0 and a 1 Some examples are: f(x) = 3 x g(x) = (1/3) x h(x) = (4.3) x Note that the variable in an exponential function is in the. Graphs of Exponential Functions f(x) = a x Increasing if a > 1 Decreasing if 0 < a < 1 Domain: (-, ) Range: (0, ) x-intercept: none y-intercept: (0, 1) Vertical asymptote: none Horizontal asymptote: y = 0 Once you know the general shape of the graph of an exponential function, you can graph transformations of the function using the techniques you learned in Section 1.7. Example 1 Example Graph the function f(x) = 3 x. State the horizontal asymptote. Graph the function f(x) = (1/3) x. State the horizontal asymptote. Example 3 Sketch the graph of the function f(x) = (1/3) x-1. Describe how the graph can be obtained from the graph of a basic exponential function. State the horizontal asymptote. Compound Interest Formula The amount of money A that a principal P will grow to after t years at interest rate r (in decimal form), compounded n times per year, is given by the formula A is the ending balance P is the principal r is the interest rate as a decimal t is the number of years n is the number of times per year that interest is compounded Example 4 Suppose that $750 is invested at 4.5% interest, compounded semiannually. a) Find the function for the amount to which the investment grows after t years. b) Find the amount of money in the account after 5 years.

5 Example 5 On Jacob s sixth birthday, his grandparents present him with a $3000 certificate of deposit (CD) that earns 3.6% interest, compounded quarterly. If the CD matures on his sixteenth birthday, what amount will be available then? 5 The Natural Number e The natural number e occurs as the base of an exponential function in various applications. You can approximate e with your calculator by letting and letting x approach in the table. This is the compound interest formula using P = 1, r = 100%, and t = 1. The value of e to nine decimal places is e Example 6 Example 7 Example 8 Find the value of e 4 to four decimal places. Find the value of e to four decimal places. Graph the function f(x) = + e -x. State the horizontal asymptote.

6 Section 5.3 Logarithmic Functions and Graphs 6 Logarithmic Function Logarithmic functions are of exponential functions. Consider f(x) = x. Find the inverse function. y = log a x if and only if x = a y, where x > 0, a > 0, and a 1. We read log a x as the logarithm, base a, of x. A logarithm is an. Evaluating Logarithms without a Calculator Example 1 a) log 16 Find each of the following without using a calculator. b) log c) log 16 4 d) log e) log a 1 f) log a a Converting Between Exponential and Logarithmic Equations Example a) 5 = 5 x Convert each of the following to a logarithmic equation. b) e w = 30 Example 3 Convert each of the following to an exponential equation. a) log = 3 b) log b R = 1

7 Common Logarithm log x means log 10 x 7 Common logarithms were used to solve complicated calculations before calculators became so widely available. Example 4 Find each of the following logarithms on a calculator, rounding to four decimal places. a) log 73,456 b) log c) log (-4) Natural Logarithm ln x means log e x Example 5 Find each of the following logarithms on a calculator, rounding to four decimal places. a) ln 73,456 b) ln c) ln (-4) d) ln 1 e) ln e The Change of Base Formula For any logarithmic bases a and b, and any positive number M, log log b M a M loga b Example 6 Find the following logarithms using the change-of-base formula and your calculator, rounding to four decimal places. a) log 6 8 b) log

8 Graphs of Logarithmic Functions f(x) = log a x 8 Domain: (0, ) Range: (-, ) x-intercept: (1, 0) y-intercept: none Vertical Asymptote: x = 0 Horizontal Asymptote: none One way to graph a logarithmic function is to use the change of base formula to convert the logarithm to either base 10 or base e and complete the graph as in the previous section with exponential functions. Example 7 Graph the function f ( x) 3 log x. Describe how the graph can be obtained from the graph of a basic logarithmic function. Give the domain and the vertical asymptote. Example 8 Graph the function f ( x) ln( x ). Describe how the graph can be obtained from the graph of a basic logarithmic function. Give the domain and the vertical asymptote. Applications of Logarithmic Functions Example 9 In a study by psychologists Bornstein and Bornstein, it was found that the average walking speed w, in feet per second, of a person living in a city of population P, in thousands, is given by the function w(p) = 0.37 ln P The population of Philadelphia, Pennsylvania, is 1,517,600. Find the average walking speed of people living in Philadelphia. Round to the nearest tenth. Example 10 The ph of tomatoes is 4.. Find the hydrogen ion concentration, [H + ], of tomatoes. Use the formula ph = - log [H + ]. Express your answer in scientific notation.

9 Section 5.4 Properties of Logarithmic Functions 9 The Product Rule loga MN loga M loga N Example 1 Express as a sum of logarithms: log 0.x Example Express as a single logarithm and, if possible, simplify: log 3 x log 3 w The Power Rule loga M p ploga M Example 3 Express as a product: 5 loga 11 The Quotient Rule loga M loga M loga N N Example 4 Express as a difference of logarithms: loga b 10 Example 5 Express as a single logarithm and, if possible, simplify: logw 15 logw 5 Applying the Properties Together Example 6 Example 7 Example 8 Express in terms of sums and differences of logarithms: loga w 3 y 4 z Express in terms of sums and differences of logarithms: m 8 n 1 loga 4 a 3 b 5 Express as a single logarithm and, if possible, simplify: 1 6log b x log b y log 3 b z

10 The Logarithm of a Base to a Power 10 loga a x x A Base to a Logarithmic Power log a a x x Example 9 Simplify the following: a) loga a 6 b) ln e -8 log c) 7 7 w d) e ln 8

11 Section 5.5 Equations Solving Exponential Equations and Logarithmic 11 Exponential Equations Equations with variables in the exponents, such as 3 x = 40 and 5 3x = 5, are called exponential equations. Solving Exponential Equations using the Base-Exponent Property If both sides of the equation can easily be written using the same base, then this solution method can be used. For our purposes, the base-exponent property says that if each side of an equation can be written entirely with the same base on each side, then the exponents must equal each other. Once the bases are the same, set the equal to each other and solve for the variable. Example 1 Solve: 5 x 3 = 15 Solving Exponential Equations using the Property of Logarithmic Equality If the bases cannot easily be made the same on both sides of the equation, then use this solution method. Isolate the term with the exponent (assuming that leaves only one term on each side of the equation). If there is a negative in front of the exponential term, divide both sides by -1 to make the term positive. Take a logarithm of both sides of the equation. The natural logarithm is a good choice. Use the power rule for logarithms to drop the exponent out in front of the logarithm. Use your algebraic skills to isolate the variable. Example Solve: x = 50 Example 3 Solve: e -.5w = 1 Example 4 Solve: 5 x + = 4 1 x Logarithmic Equations Equations containing variables in logarithmic expressions, such as log x 16 and log x + log (x + 4) = 1, are called logarithmic equations.

12 Solving Logarithmic Equations (with some terms without logs) Put all of the log terms on one side of the equation and the terms without logs on the other side of the equation. 1 Use the product and quotient rules of logarithms to write a single logarithm (assuming that there is more than one log term) Change the equation to form by using the definition of a logarithm. Solve for the variable. YOU MUST CHECK YOUR ANSWERS TO MAKE SURE THAT YOU WILL NOT BE TAKING THE LOGARITHM OF A NEGATIVE NUMBER OR OF ZERO. Example 5 Solve: log 4 x 3 Example 6 Solve: log ( x 1) log ( x 1) 3 Solving Logarithmic Equations (when every term has a log) Use the product and quotient rules of logarithms to write each side of the equation with a single logarithm. Rewrite the equation without the logarithms using the property of logarithmic equality. For our purposes, this means that if the logarithms on both sides of the equation have the same base, then the arguments (expressions inside the logarithm) must be equal to each other. Solve for the variable. YOU MUST CHECK YOUR ANSWERS TO MAKE SURE THAT YOU WILL NOT BE TAKING THE LOGARITHM OF A NEGATIVE NUMBER OR OF ZERO. Example 7 Solve: log 8 ( x 1) log 8 x log 8 5

13 Solving Equations Graphically Enter the left side of the equation as Y1 and the right side as Y and adjust the viewing window to see the point(s) of intersection of the two graphs. 13 Press nd CALC and choose intersect. Make sure that both an x and y value appear at the bottom of the screen and that Y1 is at the top of the screen when asked the question First curve?. Then press ENTER. Make sure that both an x and y value appear at the bottom of the screen and that Y is at the top of the screen when asked the question Second curve?. Then press ENTER. You may move the cursor to sit on the point of intersection when asked the question Guess? Then press ENTER. The calculator will say Intersection and give you the x and y value of the intersection point. Example 8 Find approximate solution(s) of the equation to three decimal places. 8.e 0.05x = 3.4

14 Section 5.6 Applications and Models: Growth and Decay; Compound Interest 14 Basic Exponential Growth/Decay Model P(t) = P 0 e kt P 0 is the initial population when time is 0 P(t) is the population after some time t t is the amount of time k is the exponential growth/decay rate (k is positive for growth and negative for decay) Example 1 Suppose that P dollars is invested at an interest rate of 4.7% per year, compounded continuously. a) Find the exponential growth function in terms of P. b) If $1,000 is invested, what is the balance after 5 years? Round to the nearest cent. c) What is the doubling time? Round to the nearest tenth of a year. Example The amount of carbon-14 present in organic matter after t years is given by P(t) = P 0 e t. If a piece of charcoal that had lost 7.3% of its original amount of carbon was discovered from an ancient campsite, what is the approximate age of the charcoal? Round to the nearest year. Example 3 The decay rate of Krypton-85 is 6.3% per year. What is its half-life? Round to the nearest year. Example 4 The number N of farms in the United States has declined continually since In 1950, there were 5,650,000 farms, and in 004 that number had decreased to,110,000 (Sources: U.S. Department of Agriculture; National Agricultural Statistics Service). Assuming the number of farms decreased according to the exponential model: a) Find the value of k to six decimal places, and write an exponential function that describes the number of farms after time t, in years, where t is the number of years since b) Estimate the number of farms in 009. Round your answer to the nearest tenthousand. c) At this decay rate, when will only 1,000,000 farms remain?

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