Honors Advanced Algebra Chapter 8 Exponential and Logarithmic Functions and Relations Target Goals By the end of this chapter, you should be able to Graph exponential growth functions. (8.1) Graph exponential decay functions. (8.1) Solve exponential equations. (8.2) Use exponential equations to solve compound interest problems. (8.2) Evaluate logarithmic expressions. (8.3) Solve logarithmic equations. (8.4) Simplify and evaluate expressions using the properties of logarithms. (8.5) Solve logarithmic equations using the properties of logarithms. (8.5) Solve exponential equations using common logarithms. (8.6) Evaluate logarithmic expressions using the change of base formula. (8.6) Evaluate expressions involving the natural base and natural logarithm. (8.7) Solve exponential equations using natural logarithms. (8.7) Use logarithms to solve problems involving exponential growth and decay. (8.8)
Honors Advanced Algebra Chapter 8 - Exponential and Logarithmic Functions and Relations Assignment Guide 8.1 Graphing Exponential Functions Target Goals: Graph exponential growth functions Graph exponential decay functions HW #1 Worksheet #1 8.2 Solving Exponential Equations Target Goal: Solve exponential equations Use exponential equations to solve compound interest problems. HW #2 pg 488 #1-4, 6, 9-14, 20-22, 33, 35, 37 8.3 Logarithms and Logarithmic Functions Target Goal: Evaluate logarithmic expressions HW #3 Worksheet #3 8.4 Solving Logarithmic Equations Target Goal: Solve logarithmic equations HW #4 pg 504 #1-3, 8-19 QUIZ 8.1-8.3 8.5 Properties of Logarithms Target Goals: Simplify and evaluate expressions using the properties of logarithms Solve logarithmic equations using the properties of logarithms HW #5 pg 512 #8-11, 23-26, 37-49 odd 8.6 Common Logarithms Target Goals: Solve exponential equations using common logarithms Evaluate logarithmic expressions using the change of base formula HW #6 Worksheet #6 8.7 Base e and Natural Logarithms Target Goals: Evaluate expressions involving the natural base and natural logarithm Solve exponential equations using natural logarithms HW #7 pg 529 #1-4, 9-14, 21-27 odd, 35-39 odd, 47, 53, 55 QUIZ 8.4-8.6 8.8 Common Logarithms Target Goals: Use logarithms to solve problems involving exponential growth and decay. HW #8 Worksheet 8.8 #1-5 (Day 1) Worksheet 8.8 #6-9 (Day 2) Chapter 8 Review HW #9 Practice Test QUIZ 8.8 Chapter 8 Review HW #10 pg 542 #11, 18-21, 25-28, 31-32, 35-37, 44-47, 49-52, 55, 56-58, 62 Tentative Chapter 8 Test Date: WEDNESDAY, FEBRUARY 5th
Honors Advanced Algebra Name: S8.1 Graphing Exponential Functions Date: Notes Period: Target Goals: Graph exponential growth functions. Graph exponential decay functions. A function like f ( x)= b x, where the base is a constant and the exponent contains the independent variable, is an exponential function. Examples:, 10 2 x 3, and 1000e 0.08t 4 x GRAPHING FUNDAMENTAL EXPONENTIAL FUNCTIONS The graph of the fundamental exponential function f ( x)= b x is continuous and asymptotic in one extreme of the domain. When b > 1, the function is increasing and known as an exponential growth function. The function f ( x)= b x contains key points ( 0,1), ( 1,b ), and 1, 1 b which are plotted to assist graphing the function by hand. Graph the exponential growth function using three points. Ex 1. y = 2 x Ex 2. f ( x)= 4 x x y x y When b < 1, the function is decreasing and known as an exponential decay function. Graph the exponential decay function using three points. Ex 3. y = 1 3 x Ex 4. f ( x )= 4 5 x x y x y
Ex 5. Which of the following is the graph of y = 2 x 4? Ex 6. Which of the following is the graph of 2 y = 4 x + 3? Assignment #1: Worksheet #1
Honors Advanced Algebra Name: S8.2 Solving Exponential Equations Date: Notes Period: WarmUp: Rewrite each number as a power with the smallest base possible. Negative exponents are acceptable. 1 1. 9 2. 8 3. 0.1 4. 49 Target Goal: Solve exponential equations. Use exponential equations to solve compound interest problems. In an exponential equation, variables occur as exponents. PROPERTY OF EQUALITY FOR EXPONENTIAL FUNCTIONS Let b > 0 and b 1. b x = b y if and only if x = y. Example: If 3 x = 3 5, then x = 5. If x = 5, then 3 x = 3 5. The Equality Property can be used to solve exponential equations if the exponential expressions can be written with the same (common) base. Solve the exponential equation using the Equality Property. Ex 1. 3 x = 9 4 Ex 2. 8 5 x = 4 2 x 1 Ex 3. 5 2 x 3 = 1 25 Ex 4. 1 2 4x+ 1 = 8 2x+ 1 Ex 5. 5x+ 1 x 4 2 27 = 3 8
COMPOUND INTEREST You can calculate compound interest using the exponential function: A = P 1+ r n nt where A is the amount in the account after t years, P is the principal amount invested, r is the annual interest rate, and n is the number of compounding periods each year. Solve the compound interest problem. Ex 4. An investment account pays 4.2% annual interest compounded monthly. If $2500 is invested in this account, what will the balance be after 15 years? You try it! 1. 2 5 x 6 = 8 x 2. 8 x+5 = 16 2 x 3. 7 5 x = 1 343 4. A money market account yields 1.8% annual interest compounded quarterly. If $4500 is invested, what will the balance be after 5 years? Assignment #2: page 488 #1-4, 6, 9-14, 20-22, 33, 35, 37
Honors Advanced Algebra Name: S8.3 Logarithms and Logarithmic Functions Date: Notes Period: WarmUp: Solve. 1. 4 3x = 8 x+5 2. 6 2 x = 1 216 Target Goal: Evaluate logarithmic expressions. In the equation x = b y, the variable y is called the logarithm of is read y equals log base b of x. x. This is usually written as y = log b x, which LOGARITHM WITH BASE b Let b and x be positive numbers, b 1. The logarithm of x with base b is denoted log b x and is defined as the exponent y that makes the equation b y = x true. For x > 0, there is a number y such that Example: If log 3 27 = y, then 3 y = 27. log b x = y if and only if b y = x (logarithmic form) (exponential form) It is often helpful to rewrite a logarithmic equation into exponential form or an exponential equation into logarithmic form. Write the equation in exponential form. Ex 1. log 3 9 = 2 Ex 2. log 10 1 100 = 2 Write the equation in logarithmic form. 1 Ex 3. 5 3 = 125 Ex 4. 27 3 = 3
Evaluate. Ex 5. log 3 243 Ex 6. log 7 1 49 Ex 7. log 1 3 27 Solve for x. Ex 8. 1 log x = 3 Ex 9. log 4( x 3) = 2 Ex 7. log 2 8 = 4x + 5 64 You try it! 3 1. Write log 1 125 = 3 in exponential form. 2. Write 4 2 = 8 in logarithmic form. 5 1 3. Evaluate log 8 16. 4. Solve. log 3(2x 5) = 3 Assignment #3: Worksheet #3
Honors Advanced Algebra Name: S8.4 Solving Logarithmic Equations Date: Notes Period: WarmUp: 1. Rewrite in exponential form. 2. Rewrite in logarithmic form. 3. Evaluate. log 25 5 = 1 2 4 2 = 1 16 log 3 81 Target Goal: Solve logarithmic equations. A logarithmic equation contains one or more logarithms. One way to solve a logarithmic equation involves rewriting the equation into exponential form. Solve the logarithmic equation. Ex 1. log 8 x = 4 3 Ex 2. log 16 x = 5 2 Ex 3. log x 8 = 3 2 Ex 4. log 4 1 2 = x PROPERTY OF EQUALITY FOR LOGARITHMIC FUNCTIONS Let b > 0 and b 1. log b x = log b y if and only if x = y. Example: If log 5 x = log 5 8, then x = 8. If x = 8, then log 5 x = log 5 8. The Equality Property can be used to solve logarithmic equations if the logarithmic expressions have the same (common) base.
Solve the logarithmic equation using the Equality Property. Ex 5. log 9 2x = log 9 ( 24 x) Ex 6. log 5 x 2 = log 5 x + 6 ( ) Ex 7. log 2 ( x 2 + 6x)= log 2 x 4 ( ) You try it! 1. log 5 x = 2 2. log 3 2a = 2 3. log 2 x 16 = 2 4. log 9 ( 5x 1)= log 9 ( 3x + 7) 5. log 6 ( x 3)= log 6 ( 2x) Assignment #4: page 504 #1-3, 8-19
Honors Advanced Algebra Name: S8.5 Properties of Logarithms Date: Notes Period: WarmUp: Solve. 1. log 5 x = 3 2. log 7 x + 2 ( )= log 7 4x Target Goals: Simplify and evaluate expressions using the properties of logarithms. Solve logarithmic equations using the properties of logarithms. Since logarithms are exponents, the properties of logarithms can be derived from the properties of exponents. PRODUCT PROPERTY OF LOGARITHMS The logarithm of a product is the sum of the logarithms of its factors. For all positive numbers a, b, and n, where n 1, log n ab = log n a + log n b ( )( 6) Example: log 2 5 = log 5 + log 6 2 2 You can use the Product Property to solve some logarithmic equations. Solve the logarithmic equation using the Product Property. Ex 1. log 4 x = log 4 3 + log 4 5 Ex 2. log 5 ( y 4)+ log 5 ( y + 4)= log 5 84 Ex 3. log 10 x + log 10 ( x + 9)= 1 QUOTIENT PROPERTY OF LOGARITHMS The logarithm of a quotient is the difference of the logarithms of the numerator and denominator. For all positive numbers a, b, and n, where n 1, log n a b = log n a log n b Example: log 2 5 6 = log 2 5 log 2 6
You can use the Quotient Property to solve some logarithmic equations. Solve the logarithmic equation using the Quotient Property. Ex 4. log 8 x log 8 3 = log 8 5 Ex 5. log 3 4 = log 3 6 log 3 x POWER PROPERTY OF LOGARITHMS The logarithm of a power is the product of the logarithm and the exponent. For any real number p, and positive numbers m and b, where b 1, log b m p = plog b m Example: log 2 6 5 = 5log 2 6 You can use the Power Property to solve some logarithmic equations. Solve the logarithmic equation using the Power Property. Ex 6. 2log 4 x = log 4 9 Ex 7. 4 log 7 x = log 7 1 81 You try it! 1. log 10 7 + log 10 ( n 2)= log 10 6n 2. 3log 7 x = log 7 27 3. log 10 ( m + 3) log 10 m = log 10 4 4. log 9 x = 1 2 log 144 1 9 3 log 8 9 Assignment #5: page 512 #8-11, 23-26, 37-49 odd
Honors Advanced Algebra Name: S8.6 Common Logarithms Date: Notes Period: WarmUp: Solve. 1. log 12 x 2 ( )+ log 12 x + 3 ( )= log 12 6 2. 3log 5 2 log 5 x = log 5 4 Target Goals: Solve exponential equations using common logarithms. Evaluate logarithmic expressions using the change of base formula. Base 10 logarithms are called common logarithms. Common logarithms are usually written without the subscript 10. log x = log 10 x, x > 0 If both sides of an exponential equation cannot easily be written as powers of the same (common) base, one can solve the equation by taking the logarithm of each side. Solve the exponential equation using logarithms. Ex 1. 5 x = 62 Ex 2. 7 x = 20 Ex 3. 3 x+1 = 2 3x Ex 4. 3 x 4 = 5 x 1
CHANGE OF BASE FORMULA For all positive numbers a, b, and n, where a 1 and b 1, log a n = log b n log b a Example: log 3 11 = log 10 11 log 10 3 = log11 log 3 The change of base formula makes it possible to evaluate a logarithmic expression of any base by rewriting the expression into one that involves common logarithms. Evaluate using the change of base formula. Ex 5. log 4 23 Ex 6. log 14 2 3 You try it! 1. 25 x2 = 50 2. 5 4 y+1 = 32 2 y 3. log 15 5 4. log 11 104 Assignment #6: Worksheet #6
Honors Advanced Algebra Name: S8.7 Base e and Natural Logarithms Date: Notes Period: WarmUp: 1. Solve 3 x = 14 2. Evaluate log 6 83 Target Goals: Evaluate expressions involving the natural base and natural logarithm. Solve exponential equations using natural logarithms. Like π and 2, the number e is an irrational number. The value is 2.71828... It is referred to as the natural base, e. An exponential function with base e is called a natural base exponential function. NATURAL BASE FUNCTIONS The function f ( x)= e x is used to model continuous exponential growth. The function f ( x)= e x is used to model continuous exponential decay. The inverse of a natural base exponential function is called the natural logarithm. This logarithm can be written as log e x, but is more often abbreviated as ln x. You can write a natural base exponential equation for a natural logarithmic equation: ln 4 = x log e 4 = x e x = 4 Write the natural logarithmic equation in exponential form. Ex 1. ln x = 0.5381 Ex 2. ln25 = x Write the natural base exponential equation in logarithmic form. Ex 3. e x = 23 Ex 4. e 4 = x
Equations involving base e are easier to solve using natural logarithms rather than common logarithms. Solve. Round to the nearest ten-thousandth (4 decimal places). Ex 5. 3e 2 x + 4 = 10 Ex 6. 5e 3x 2 = 6 Solve. Round to the nearest ten-thousandth (4 decimal places). Ex 7. 2 ln 5x = 6 Ex 8. 4 ln 3x = 11 CONTINUOUSLY COMPOUNDED INTEREST You can calculate continuously compounded interest using the natural base exponential function: A = Pe rt where A is the amount in the account after t years, P is the principal amount invested, and r is the annual interest rate. Solve the continuously compounded interest problem. Ex 9. An investment account pays 4.2% annual interest compounded continuously. a. If $2500 is invested in this account, what will the balance be after 15 years? b. How long will it take for any investment to double in value? Assignment #7: page 529 #1-4, 9-14, 21-27 odd, 35-39 odd, 47, 53, 55
Honors Advanced Algebra Name: S8.8 Using Exponential Functions Date: Notes Period: Target Goals: Use logarithms to solve problems involving exponential growth and decay. Recall: CONTINUOUSLY COMPOUNDED INTEREST You can calculate continuously compounded interest using the natural base exponential function: A = Pe rt where A is the amount in the account after t years, P is the principal amount invested, and r is the annual interest rate. Ex 1. Carl hopes to invest $500 at 8.25% annual interest, compounded continuously. How long would it take for his money to triple? NATURAL EXPONENTIAL GROWTH AND DECAY You can model natural exponential growth and decay in a similar way using: y kt = ye 0 or y e y = 0 kt where y is the amount after t units of time, y o is the initial amount (or the amount when t = 0, thus the subscript of 0), and k is the growth or decay constant. The constant k is positive with natural exponential growth and negative with natural exponential decay. The half-life of a decay model is the length of time t1 2 that it takes for y to decay to one-half of y o. or y 1 y = 2. 0 Ex 2. A certain strain of bacteria grows from 40 to 326 in 120 minutes. Find k for its growth model.
Ex 3. A certain radioactive isotope decays from 100 grams to 15 grams in 13.2 years. Find its decay constant and half-life. EXPONENTIAL APPLICATIONS PACKET Let s look together at #4 in the packet. What s x and what s y? Let s look together at #7 in the packet. Set up both ordered pairs. Let s set up and complete #1 in the packet. Assignment #8: Day 1-8.8 Exponential Applications Packet #1-5 Day 2-8.8 Exponential Applications Packet #6-9