5.2 (Version B) Math 112 Section: Instructor's name: Name: 1. Consider the transformation of the natural exponential = +2 + 4. Describe the transformations in an appropriate order. Draw a sketch and use it to determine the domain, range, and asymptote of = +2 + 4. Main Algebraic Solution: Check Your Solution: Section 5.2 (Version B) Math 112 Section: Instructor's name: Name: 2. Samantha wants to have $20,000 in 15 years. How much does she have to invest now if interest is paid at a rate of 4.2% compounded continuously? Round to the nearest $0.01. Main Algebraic Solution: Check Your Solution: Section 5.3 (Version B) Math 112 Section: Instructor's name: Name: 1. Consider the transformation = log 2 ( + 4) 1. Describe the transformations in an appropriate order. Draw a sketch and use this to determine the domain, range, and asymptote of = log 2 ( + 4) 1. Make sure you check all of the features of the function. Main Algebraic Solution: Check Your Solution: Section 5.3 (Version B) Math 112 Section: Instructor's name: Name: 2. Find the inverse of = log 2 ( + 4) 1. Main Algebraic Solution: Check Your Solution: Section 5.4 (Version B) Math 112 Section: Instructor's name: Name: 1. Use properties of logarithms to completely expand the logarithmic expression. Wherever possible, evaluate logarithmic expressions. Main Algebraic Solution: log 5 ( 252 ) Check Your Solution: Section 5.4 (Version B) Math 112 Section: Instructor's name: 2. Use properties of logarithms to rewrite as a single logarithm. 1 4 log 7 () log 7 () + log 7 () 7log 7 () 5 Main Algebraic Solution: Check Your Solution: Name: Section 5.2 - The Natural Exponential Function Objectives Identify the characteristics of the natural exponential function () = , including the domain, range, intercept, asymptote, end behavior, and its graph. Sketch the graph of natural exponential functions using transformations. Solve natural exponential equations by relating the bases. Solve continuous compound interest application problems. Determine the present value of an investment using continuous compound interest. Solve population growth application problems. Preliminaries Consider the natural exponential function () = . List the following properties. Domain: Range: y-intercept: Asymptote: End behavior: and Graph: Continuous compound interest can be calculated by using the formula = Write down the meaning of each value in the formula. : : : : Page | 204 Warm-up 6. Use your calculator to approximate the following values rounded to four decimal places. (A) 3 (B) 12 0.16 2. Determine the transformations that are performed on a base function. (A) () = 4( + 2)3 (B) () = 3 5 Page | 205 Class Notes and Examples 5.2.1 Each of the following functions were created using transformations of () = . Determine the transformations that were performed. List the domain, horizontal asymptote, range, and y-intercept. Graph the given function. (Note: we will discuss an algebraic method to determine the x-intercept later in chapter 5.) (A) () = + 2 Transformation(s): Domain: Horizontal Asymptote: Range: y-intercept: Graph: Page | 206 (B) () = 2 3 Transformation(s): Domain: Horizontal Asymptote: Range: y-intercept: Graph: Page | 207 5.2.2 Solve the following equations. Check your answers in the original equation. 3 (A) 5 = 1 (B) 2 3 = +1 5.2.3 Dmitry invests $3200 in a savings account that earns 4.6% interest compounded continuously. How much money would Dmitry have in the account after 3.5 years? 5.2.4 Anna has a choice between two investment options for a $1000 gift she received. The first option earns 7.8% interest compounded continuously. The second option earns 7.9% interest compounded semi-annually. Which option would yield the greatest amount of money after 5 years? Page | 208 5.2.5 Arturo wants to have $15,000 in 6 years, so he will place money into a savings account that pays 3.7 % interest compounded continuously. How much should Arturo invest now to have $15,000 in 6 years? Check your answer. What is the general exponential growth model? 5.2.6 The population of a city can be measured by () = 12,500 0.02 , where t represents time in years after 1985. (A) What was the population in 1985? (B) What was the population in 2000? (C) What does the model predict the population to be in the year 2020? Page | 209 5.2.7 An invasive beetle was discovered in a small Pacific island 15 years ago. It is estimated that there are 12,400 beetles on the island now, with a relative growth rate of 16%. (A) How many beetles were initially discovered 15 years ago? (B) How many beetles will there be after another 15 years? Page | 210 Section 5.2 Self-Assessment (Answers on page 257) 1. (Multiple Choice) What is the range of = + 12? (A) (0, ) (B) (, 0) (D) (12, ) (E) (, 12] (C) (, 12) 2. Kathryn invested $7500 in an investment account that earned 6.1% interest compounded continuously for 40 years. How much money is in the account after 40 years? 3. (Multiple Choice) Marco needs to have $1,000,000 in an investment account in 35 years. How much should be invested today at an interest rate of 8.2% compounded continuously to have $1,000,000 in 35 years? The minimum amount that should be invested is: (A) (B) (C) (D) (E) More than $68,000 Between $65,000 and $68,000 Between $62,000 and $65,000 Between $59,000 and $62,000 Between $56,000 and $59,000 4. Anna has a choice between two investment options for a $1000 gift she received. The first option earns 7.8% interest compounded continuously. The second option earns 7.9% interest compounded semi-annually. Calculate the amount of money earned in each investment and determine which option would yield the greatest amount of money after 5 years. 5. (Multiple Choice) A species of owl was introduced in an area 30 years ago. It is estimated that there are 6700 owls in the area now, and the population has a relative exponential growth rate of 6% per year. How many owls will there be 25 years from now? (A) (B) (C) (D) (E) Less than 29,000 owls Between 29,000 owls and 31,000 owls Between 31,000 owls and 33,000 owls Between 33,000 owls and 35,000 owls More than 35,000 owls Page | 211 Section 5.3 - Logarithmic Functions Objectives Apply the definition of a logarithm to convert an equation in logarithmic form into exponential form. Given an equation in exponential form, rewrite the equation in logarithmic form. Properly identify the notation for natural (base e) and common (base 10) logarithm. Evaluate natural and common logarithms on a calculator. Know and identify the shape and basic features of the graph of () = ln and () = log . Evaluate simple logarithms of any allowable base without the use of a calculator. Determine the domain of logarithmic functions. Preliminaries Define in your own words what it means for a function to be the inverse of another function. (see section 3.6 if you need a little review.) If (1,2) is on the graph of = (), what point must be on the graph of = 1 ()? (see section 3.6 if you need a little review.) Rewrite the equation in exponential form. = log is equivalent to Rewrite the equation in logarithmic form. = is equivalent to Write the cancellation properties of exponentials and logarithms. Page | 212 Warm-up 1. Consider the function () = 2 . Answer the following questions (A) Sketch an accurate graph of = (). Sketch an accurate graph of = 1 () Label coordinates for two points. Label coordinates for two points. (B) State the domain of . State the domain of 1 . (C) State the range of . State the range of 1 . (D) State the asymptote(s), if any. State the asymptote(s), if any. (E) Explain why () = 2 has an inverse function. Page | 213 Class Notes and Examples Definition: For > 0, > 0, and 1, the logarithmic function with base is defined by = log if and only if = . 5.3.1 Change each exponential equation into logarithmic form. 1 (A) 23 = 8 (B) 32 = 9 (C) 161/2 = 4 5.3.2 Change each logarithmic equation into exponential form. (A) log 2 32 = 5 (B) log1/3 27 = 3 5.3.3 Evaluate each expression without using a calculator. Verify your answer by evaluating the equivalent exponential expression. 1 (A) log 3 9 (B) log 2 (C) log 4 2 8 Page | 214 Notation: Common logarithm: log10 () is denoted simply as log() Natural logarithm: log () is denoted simply as ln() Useful for approximating values on a calculator. What are the basic properties of logarithms? (Assume > 0 and 1.) log = log 1 = log = log ( ) = 5.3.4 Use the basic properties of logarithms to simplify the following expressions without using your calculator. (A) log 3 (312 ) (B) log 1000 (C) 5log5 (11) Page | 215 5.3.5 Consider the function () = log 2 . Note that () is the inverse of the function () = 2 described in the warm-up section. Use the properties of inverse functions and the warm-up exercise to answer the following. (A) Sketch an accurate graph of (), labeling coordinates for two points. (B) State the domain of (). (C) State the range of (). (D) State the asymptote(s) of (), if any. Page | 216 5.3.6 Summarize the properties of the graphs of logarithmic functions with different bases. = log for > 1 = log for 0 < < 1 Domain: Domain: Range: Range: x-intercept: x-intercept: Asymptote: Asymptote: Increasing or Decreasing? Increasing or Decreasing? Graph: Graph: Page | 217 How do we determine the domain of logarithms? 5.3.7 Find the domain of each logarithmic function. (A) () = log(3 2) (B) () = log 5 (4 + 7) (C) () = ln( 2 4) Page | 218 5.3.8 For each logarithm function below, determine the domain, intercepts, and the equation of the asymptote. Then sketch the graph of each function, labeling the exact coordinates of at least two points. (A) () = ln( 1) + 3 Page | 219 (B) () = log 4(3 + 2) Page | 220 Section 5.3 Self-Assessment (Answers on page 257) 1. (Multiple Choice) Write in logarithmic form. 26 = 3 2. (A) log (26) = 3 (B) log 3 (26) = (D) log 26 (3) = (E) log 26 () = 3 (C) log (3) = 26 (Multiple Choice) Write in exponential form. log (6) = 37 (A) 6 = 37 (B) 376 = (D) 6 = 37 (E) 37 = 6 (C) 37 = 6 3. Determine the domain of the logarithmic function () = log 3 (8 + 5). 4. Determine the x-intercept(s), if any, of () = log 4 (3 + 11). Page | 221 Section 5.4 - Properties of Logarithms Objectives Apply the properties of logarithms to expand expressions involving the logarithm of a product/quotient/power into a sum/difference of logarithms. Apply the properties of logarithms to condense a sun/difference of logarithms into a single logarithm. Use the change of base formula to convert a logarithm with any allowable base to a logarithm with any allowable base. (In particular, convert to base 10 or base e.) Use the one-to-one property of logarithms to solve certain logarithmic equations. Preliminaries Carefully write the Product, Quotient, and Power Rules for logarithms. Write the change of base formula Page | 222 Warm-up Evaluate the following, if they are defined, without a calculator: 1 1. log(10,000) 2. log 5 (517 ) 3. log1/2(16) 4. log 3 (0) Page | 223 Class Notes and Examples 5.4.1 Use properties of logarithms to expand each expression as much as possible. (A) log ( 2 3 ) = 1 (B) ln () = (C) log(5) = Page | 224 (D) log(101/3 ) = (E) log 2 ( (F) ln (3 ) = 2 ) = 3 Page | 225 5.4.2 Use properties of logarithms to rewrite the following as a single logarithm. 1 (A) ln(6) + ln() ln(2) = 2 (B) log(5) log() 3 log(3) + log() = (C) ln() 2 ln() ln() = 1 (D) 2 log 2 () + log 2 () 4 log 2 () 3 log 2 () + log 2 () = Page | 226 What is the change of base formula, and when would you want to use it? 5.4.3 Use your calculator to approximate the following to four decimal places. (A) log 3 (6) (B) log 7(9) 5.4.4 Use change of base formula and properties of logarithms to rewrite the expression log 2 () + log 4 () as a single logarithm. Page | 227 What does the logarithm property of equality say? Under what circumstances can we use the logarithm property of equality to solve equations? 5.4.5 Solve the following equations. Check your solutions. (A) log(10) = log( 2 ) Page | 228 (B) log 2 (5) = log 2 () log 2 ( + 1) (C) ln() = log 2 (13) (Hint: use the change of base formula) Page | 229 Section 5.4 Self-Assessment (Answers on page 257) 1. (Multiple Choice) Write as a sum/difference and/or multiple of logarithms. log ( (A) (B) (C) (D) (E) 2. 10 ) 3 10 + log() log(3) 10 + log() 3log() 10 log() 3log() 1 + log() log(3) 1 + log() 3log() Rewrite the expression as a single logarithm. 1 2 log() log() + log() 5 log() 3 3. (Multiple Choice) Write as a sum/difference and/or multiple of logarithms. 3 ln ( 5 ) 1 (B) ln() + ln (3 ) ln() ln(5) 1 (D) ln() + ln() ln() (ln())5 (A) ln() + 3 ln() ln() 5 ln() (C) ln() + 3 ln() ln() + 5 ln() 1 3 (E) None of these 4. Solve the equation. log 2 (8) = log 2 (3) log 2 ( 1) 5. (Multiple Choice) Solve the equation. log 7 (3) = log 7 () + log 7 ( 2) (A) = 3 only 5 2 (D) = only (B) = 1, 3 only (C) = 1 only (E) There are no solutions Page | 230