Independent Study Project: Chapter 4 Exponential and Logarithmic Functions

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1 Name: Date: Period: Independent Study Project: Chapter 4 Exponential and Logarithmic Functions Part I: Read each section taken from the Algebra & Trigonometry (Blitzer 2014) textbook. Fill in the blanks with the appropriate information. Part II: Complete the MathXL for each section on the appropriate day. It will only be opened until 11:59pm on the night it is assigned. Sec. 4.1 Exponential Functions (Wed., Nov. 15) The exponential function f with base b is defined by where b is a positive constant other than 1 (b > 0 and b 1) and x is any real number. Exponential functions have a base and a exponent. Provide an example for evaluating an exponential function. Provide an example for graphing an exponential function. Characteristics of Exponential Functions of the Form f(x) = b x : 1. The domain of f(x) = b x consists of and the range of f(x) = b x consists of. 2. The graphs of all exponential functions in the form f(x) = b x pass through the point. 3. If b > 1, f(x) = b x has a graph that goes to the right and is an function. 4. If 0 < b < 1, f(x) = b x has a graph that goes to the right and is a function. 5. f(x) = b x is a one-to-one and has an that is a function. 6. The graph of f(x) = b x approaches, but does not touch the. It is a.

2 The graphs of exponential functions can be translated,,,, or. The number e is defined as the value that approaches as n gets larger and larger, and is called the. The function is called the natural exponential function. We all want a wonderful life with,, and. Achieving this goal depends on understanding how money in savings accounts grow as a result of. This is computed on the original investment as well as on any accumulated interest. If compound interest is paid twice a year, we say that the interest is compounded. If compound interest is paid four times a year, we say that the interest is compounded. If the number of compounding periods increases infinitely, we call this interest. Practice Exercises e e Sec. 4.2 Logarithmic Functions (Fri., Nov. 17) No horizontal line can be drawn that intersects the graph of an exponential function at more than one point. This means that the exponential function is and has an. The of the exponential function with base b is called the. y = log b x b y = x Provide an example of changing from logarithmic form to exponential form. Provide an example of changing from exponential form to logarithmic form.

3 Basic Logarithmic Properties Involving One: 1. log b b = 1 because is the exponent to which b must be raised to obtain. (b 1 = b) 2. log b 1 = 0 because is the exponent to which b must be raised to obtain. (b 0 = 1) Inverse Properties of Logarithms: For b > 0 and b 1 log b b x = x The logarithm with base b of b raised to a power equals that. b log bx = x b raised to the logarithm with base b of a number equals that. The logarithmic function is the inverse of the exponential function, and therefore the of the exponential function. The graph of the logarithmic function is a of the graph of the exponential function about the line. Draw the graphs illustrating the relationship between the graph of an exponential function and its inverse, a logarithmic function for bases greater than 1 and for bases between 0 and 1. Also, include the horizontal and vertical asymptotes. Characteristics of Logarithmic Functions of the Form f(x) = log b x: 1. The domain of f(x) = log b x consists of and the range of f(x) = log b x consists of. 2. The graphs of all logarithmic functions of the form f(x) = log b x pass through the point. 3. If b > 1, f(x) = log b x has a graph that goes to the right and is an function. 4. If 0 < b < 1, f(x) = log b x has a graph that goes to the right and is a function. 5. The graph of f(x) = log b x approaches, but does not touch the. It is a.

4 The logarithmic function with base is called the common logarithmic function. List the Properties of Common Logarithms General Properties Common Logarithms Give an example of a real-life phenomenon that can be modeled by a logarithmic function. The logarithmic function with base is called the natural logarithmic function. It is expressed as f(x) = lnx and read. Like the domain of logarithmic functions, the domain of the natural log function is. List the Properties of Natural Logarithms General Properties Natural Logarithms Practice Exercises Write in exponential form = log = log 9 x 3. 3 = log b log = y Write in log form = = = x

5 Sec. 4.3 Properties of Logarithms (Tue., Nov. 21) When you multiply exponents with the same base, you b m b n = the exponents. The Product Rule Let b, M, and N be positive real numbers with b 1. log b (MN) = The logarithm of a product is the. Provide an example of using the product rule to expand each logarithmic expression. When you divide with the same base, you b m b n = the exponents. The Quotient Rule Let b, M, and N be positive real numbers with b 1. log M = b N The logarithm of a quotient is the. Provide an example of using the quotient rule to expand each logarithmic expression. When an exponential expression is raised to a power, you (b m ) n = the exponents. The Power Rule Let b and M be positive real numbers with b 1, and let p be any real number. log b M p = The logarithm of a number with an exponent is the. Provide an example of using the power rule to expand each logarithmic expression.

6 Properties for Expanding Logarithmic Expressions For M > 0 and N > 0: Properties for Condensing Logarithmic Expressions For M > 0 and N > 0: product and quotient rules. of logarithms must be before you can condense them using the Provide two examples of condensing logarithmic expressions. Most calculators give the values of the logarithm and logarithm; however, to find a logarithm with any other base, use the, log M = log am b. We are able to change to any other base as long as the log a b new base is a not equal to. Practice Exercises Use common logs or natural logs and a calculator to evaluate to four decimal places. 1. log log log log π 400 Sec. 4.4 Exponential and Logarithmic Equations (Tue., Nov.28) An exponential equation contains a in the exponent. You can solve them by expressing each side of the equation as a.

7 If b M = b N, then M = N. List the steps to solving exponential equations Provide an example. What happens when you can t rewrite exponential equations so that each side has the same base? Use to Solve Exponential Equations: 1. Isolate the expression. 2. Take the common log on both sides of the equation for. Take the natural log on both sides for. 3. Simplify using one of the properties: or or. 4. Solve for. Provide an example for solving exponential equations using each property.

8 A log equation is an equation that contains a equations can be expressed in the form. in a log expression. Some log Using the Definition of a Log to Solve Logarithmic Equations Always check proposed solutions of a log equation in the original equation. Exclude from the solution set any solution that produces the log of or the log of. When rewriting the log equation in the equivalent exponential form, you need a single log whose coefficient is. Provide an example of condensing logarithms into a single log by using the properties of logs. Using the One-to-One Property of Logarithms to Solve Logarithmic Equations 1. Express the equation in the form. This form involves a single log whose coefficient is on each side of the equation. 2. Use to rewrite the equation without logs: If log b M = log b N, then M = N. 3. Solve for. 4. Check proposed solutions in the. Include in the solution set only values for which and. Medical research indicates that the risk of having a car accident increases exponentially as the concentration of alcohol in the blood increases. The risk is modeled by R = 6e 12.77x, where x is the blood alcohol concentration and R, given as a percent, is the risk of having a car accident. What blood alcohol concentration corresponds to a 7% risk of a car accident? (Under FL law, possession of alcohol by an underage person can result in imprisonment, revocation of license and a public criminal record)

9 r nt The formula A = P (1 + n ) describes the accumulated value, A, of a sum of money, P, the principal after t years at annual percentage rate r (in decimal form) compounded n times a year. How long, to the nearest tenth of a year, will it take $1000 of your graduation money to grow at $3600 at 8% annual interest compounded quarterly? Practice Exercises Solve each exponential equation by expressing each side as a power of the same base and then equating exponents x = x 1 = x = 4 x+2 Solve each log equation. Be sure to reject any value of x that is not in the domain of the original log expression. Give the exact answer and use a calculator to round to two decimal places. 1. log 5 x = 3 2. lnx = 3 3. log 5 (x 7) = 2 4. log 7 (x + 2) = 2 Sec. 4.5 Exponential Growth and Decay; Modeling Data (Thur., Nov. 30) One of algebra s many applications is to predict the done with where quantities grow or decay at a rate Name two special cases of exponential growth models: and of variables. This can be proportional to their size. Exponential Growth and Decay Models The mathematical model for exponential growth or decay is given by f(t) = A 0 e kt or A = A 0 e kt. If k > 0, the function models the amount, or size, of a If k < 0, the function models the amount, or size, of a entity. entity. A 0 is t is A is k is

10 Draw the graphs of exponential growth and exponential decay. After 5715 years, a given amount of carbon-14 will have decayed to half the original amount. Find the exponential decay model for carbon-14. The exponential decay model is. k 0, t =, and A = A = A 0 e kt A 0 = 2 = e 5715k ln 1 = 2 e 5715k ln 1 = 5715k 2 k = A = A 0 e t Nothing on Earth can grow exponentially can be shown by the.. Growth is always limited, and Logistic Growth Model The mathematical model for limited logistic growth is given by f(t) = c 1+ae bt or A = a, b, and c are constants, with and. c 1+ae bt where As time increases (t ), the expression ae bt in the model approaches, and gets closer and closer to. This means that is a horizontal asymptote for the graph of the function. It represents the that A can attain.

11 In a learning theory project, psychologists discovered that f(t) = 0.8 is a model for describing the proportion of correct responses, f(t), after t learning trials. 1+e 0.2t a. Find the proportion of correct responses prior to learning trials taking place. b. Find the proportion of correct responses after 10 learning trials. c. What is the limiting size of f(t), the proportion of correct responses, as continued learning trials take place? All of the models were given; however, you can also create functions that model data by observing patterns in. Based on the pattern, you can choose which function is a good choice for modeling the data. The number that is a measure of how well the model fits data is called the represented by. A positive value means that both x- and y-values. A negative value means that as the x-values increase, the y-values. The closer it is to or, the better the model fits the data.

12 Table 4.9 on P.501 shows the percentage of U.S. men who are married or who have been married, by age. Create a scatter plot for the data. Based on the scatter plot, what type of function would be a good choice for modeling the data? Graphing utilities display exponential models in the form. What about when the models involve base e? Because of the b = e lnb. When expressing an exponential model in base e, is equivalent to. Provide an example of rewriting a model in base e.

13 Practice Exercise The exponential models describes the population of the indicated country, A, in millions, t years after India: A = e 0.008t Iraq: A = 31.5e 0.019t Japan: A = 127.3e 0.006t Russia: A = 141.9e 0.005t 1. What was the population in Japan in 2010? 2. What was the population in Iraq in 2010? 3. Which country has the greatest growth rate? By what percentage is the population of that country increasing each year? 4. Which countries have a decreasing population? By what percentage is the population of these countries decreasing each year? 5. When will India s population be 1377 million? 6. When will India s population be 1491 million? Part III: Complete MathXL Chapter 4 Quiz. This will count as a grade. It will be available for completion Nov. 27 Dec. 3 at 11:59pm. You will have one opportunity to take the quiz, so please make sure you are prepared prior to beginning the assessment.