Chapter 6: Logarithmic Functions and Systems of Equations

Transcription

1 QUIZ AND TEST INFORMATION: The material in this chapter is on Quiz 6 and the final exam. You should complete all three attempts of Quiz 6 before taking the final exam. TEXT INFORMATION: The material in this chapter corresponds to the following sections of your text book: 4.4, 4.5, 4.6, 4.7, 4.8, 5.1, and 5.2. Please read these sections and complete the assigned homework from the text that is given on the last page of the course syllabus. LAB INFORMATION: Material from these sections is used in the following labs: Logarithmic Functions. Logarithmic Functions Systems of Equations Comprehensive Final Assignments and Lab Logarithmic Functions Quiz 6 Test Comprehensive Final 176

2 Section 1: Logarithmic Functions Definition: We now define a function that is the inverse function to an exponential function. If a (the b ) is greater than and not equal to, then y = if and only if. Example 1: What is log 2 8? Definition: The natural log is the inverse function of. Thus, the natural log is log base. We use the following special notation for the natural log. The common log is the inverse function of. Thus, the common log is log base. We use the following notation for the common log. Example 2: Convert the following statements to logarithmic form. 1. y = 2 x 2. 8 =

3 Example 3: Convert the following to exponential form. log 3 9 Example 4: Convert the following to exponential expressions and evaluate if possible. 1. log log log 3 4. log 0 5. log a 1 178

4 6. log a a 7. log a a x Example 5: Sketch the graph of f(x) = 2 x, g(x) = log 2 x and y = x on the same set of axes. Label the graphs. Properties of Graphs of Logarithmic Functions: For f(x)=log a x where a > 1, the following are properties of the graph of f(x): 1. The function is i. 2. The graph includes the points (1, ) and (,1). 3. As x approaches infinity, f(x) approaches. 4. As x approaches zero, f(x) approaches. 5. Domain = 6. Range = 179

5 Example 6: Sketch the graphs of f(x) = (1/2) x and g(x) = log (1/2) x on the same set of axes. Label the graphs. Properties of Graphs of Logarithmic Functions: For f(x) = log a x where 0 < a < 1, the following are properties of the graph of f(x): 1. The function is d. 2. The graph includes the points (1, ) and (,1). 3. As x approaches infinity, f(x) approaches. 4. As x approaches zero, f(x) approaches. 5. Domain = 6. Range = Example 7: Convert e x = 21 to logarithmic form. 180

6 Example 8: Graph y = lnx, y = -lnx, and y = ln(-x) on the same set of axes and label the graphs. Give the domain, range, and vertical asymptote of each. y = lnx Domain: Range: Vertical asymptote: y = -lnx Domain: Range: Vertical asymptote: y = ln(-x) Domain: Range: Vertical asymptote: Example 9: Find the domain of the function y = log 2 (1 - x). 181

7 Properties of logarithm: 1. log a 1 = 2. log a a = 3. log a a r = 4. log a MN = 5. log a (M/N) = 6. log a M r = Example 10: Expand the following using the log properties. 1. log 3 (5x) 2. log 3 (3x 2 ) log x 1 a x 4. log 3 a x 1 182

8 Example 11: Solve y = log 3 5. Change of Base Formula: If y = log a x, then y = log log b b x a Example 12: Use a calculator and the base conversion formula to find the logarithm, correct to three decimal places. log Example 13: Solve each of the following equations. 1. log 3 (3x - 2) = 2 183

9 2. -2 log 4 x = log log 3 (x + 4) - log 3 9 = x = 1/5 5. x 4 2 x = 2 184

10 6. 3 x = log(4x) = log5 + log(x - 1) 8. log 3 (6x + 4) = log 3 (6x + 7) Other examples and notes: 185

11 Section 2: Applications Definition: interest is interest paid on the amount borrowed or invested (does not include any interest that accrues). interest is applied yearly and is a form of linear growth. Definition: interest is interest paid on the amount borrowed and previously owed interest. interest can be applied annually, semiannually, quarterly, monthly, daily, etc. and is a form of exponential growth. Simple Interest Formula: I = Prt 1. P = 2. r = 3. t = 4. I = Compound Interest Formula: A = P 1 r nt + n 1. P = 2. r = 3. t = 4. A = 5. n = 186

12 Example 1: Let P = $100, t = 1, and r = 3%. What is the amount if compounded quarterly? What is the amount if compounded monthly? Compounded quarterly: Compounded monthly: Continuous Compounding Formula: A = Pe rt 1. A = 2. P = 3. r = 4. t = Example 2: What is the amount if compounded continuously when P = $100, t = 1, and r = 3%? 187

13 Definition: The Present Value of A dollars to be received at a future date is the principal you would need to invest now so that it would grow to A dollars in a specified time period. Example 3: Find the P needed to get $800 after 3.5 years at 7% interest compounded monthly. 188

14 Example 4: Eddie s son will be going to college in 5 years. If Eddie needs to save $30,000 over the next 5 years for his son to go to college, how much should he invest now if he will get a 3.5% interest rate compounded continuously? 189

15 Example 5: How long will it take to double an investment if r = 7% and it is compounded continuously? 190

16 Law of Uninhibited Growth and Decay: If a quantity is changing exponentially, then A = A 0 e kt where 1. A = 2. A 0 = 3. t = 4. k = (> 0 if growing, and < 0 if decaying). Example 6: A colony of bacteria increases according to the law of uninhibited growth. If the number of bacteria doubles in 3 hours, how long will it take for the size of the colony to triple? 191

17 Example 7: The half life of carbon 14 is 5600 years. If 10g are present now, how much will be present in 100 years? Other examples and notes: 192

18 Section 3: Systems of Linear Equations Definition: A is a collection of two or more equations in two or more variables. A to a system of equations is a set of values for the variables that make each equation in the system true. If you are asked to a system of equation, then you should find solutions to the system. Example 1: Is (1, 2) a solution to the following system of equations? 2x y = 0 2x ½ y = 1 Definition: When a system of equations has at least one solution, the system is said to be. Otherwise, the system is called. Definition: An equation in n variables is said to be linear, if it is equivalent to an equation of the form. Definition: If every equation in a system of equations is linear, then we call the system of equations. 193

19 Note: The simplest system of equations to study is a system of two linear equations in two variables. In this case, the graph of each equation will be a. Then there are three possible things that could happen. 1. The lines intersect. In this case, the system will have solution given by the point of. We call the equations. 2. The lines are parallel. In this case, the system will have. 3. The lines are (meaning they are the same line). In this case, there are solutions. We call the equations. Example 2: Solve the following system of equations by substitution. x + 2y = 5 x + y = 3 Example 3: Solve the following system of equations by substitution. 2x + y = 5 4x +2y = 8 194

20 Example 4: Solve the following system of equations by elimination. x + y = 5 2x y = 4 Example 5: Solve the following system of equations by elimination. x + 3y = 5 2x - y = 3 195

21 Example 6: Solve the following system of equations by elimination. 2x + y = 4-6x - 3y = -12 Note: Now let s look at systems of equations with three equations in three variables. As above, the systems could have exactly solution, solutions, or solutions. 196

22 Example 7: Use the method of elimination to solve the following system of equations. x + y z = 2 2x + y - 3z = 1 x - 2y + 4z = 5 197

23 Example 8: The perimeter of a field is 300 feet. Find the dimensions of the field if the length is half the width. Other examples and notes: 198

24 Section 4: Matrices Definition: A is a rectangular array of numbers. Each entry in the matrix has two indices: a index and a index. Example 1: Write some examples of matrices. Look at entries and determine their row index and column index. Example 2: Write what a general matrix looks like. 199

25 Note: If we write down a system of equations, but don t write down the, it is a matrix. Note that if a variable is missing in an equation in a system, then its coefficient is. Such a matrix is called an matrix. Example 3: Write a system of equations and its augmented matrix. Definition: Row operations are used to a system of equations when we write the system as an augmented matrix. The following are row operations. 1. interchanging any two rows 2. replacing a row by any nonzero multiple of that row 3. replacing a row by the sum of that row and a constant nonzero multiple of another row 200

26 Example 4: Write some matrices and perform row operations on them. 201

27 Definition: A matrix is in when the following are true about the matrix. 1. The a 11 entry is 1 and all entries below it are The first nonzero entry in each row after the first row is a 1, zeros appear below it, and it appears to the right of the first nonzero entry in any row above. 3. Any row containing all zeros, except possibly in the last entry, appears at the bottom. Example 5: Write some examples of matrices in row echelon form and some examples of matrices that are not in row echelon form. In row echelon form: Not in row echelon form: 202

28 Note: To solve a system of equations using matrices, write the augmented matrix that corresponds to the system. Use to put this matrix in row echelon form. Analyze the system of equations that corresponds to this new matrix to the original system. Example 6: Solve the following system of equations using matrices. 2x + 2y = 6 x + y + z = 1 3x + 4y z =

29 Other examples and notes: 204