7-1. Exploring Exponential Models. Vocabulary. Review. Vocabulary Builder. Use Your Vocabulary. 1. Cross out the expressions that are NOT powers.

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1 7-1 Eploring Eponential Models Vocabular Review 1. Cross out the epressions that are NOT powers. 16 6a 1 7. Circle the eponents in the epressions below a z Vocabular Builder eponential deca (noun) ek spoh NEN shul dee KAY Related Word: deca (verb) Definition: For the function 5 ab, if a. 0 and 0, b, 1, the function represents eponential deca. Math Usage: The equation 5 (0.98) represents eponential deca. Main Idea: Something that decreases as time passes is decaing. Eample: A car loses about 15% of its value ever ear. Its value decas eponentiall. Use Your Vocabular. Circle the equations that represent eponential deca. 5(0.1) 5 (0.) 5 Q R 5 Q 7 5 R P 5 P 0 (1.05) n N 5 N 0 (0.5) t ? ? n. Write numbers to make each equation represent eponential deca. Answers ma var. Samples are given. 5 Q R n m a b Q 5 Q 0 a b t Chapter 7 186

2 Problem 1 Graphing an Eponential Function Got It? What is the graph of 5? 5. Complete the following table of values. 6. Graph the points from our table. Then connect the points with a smooth curve O 16 Summar Eponential Functions For the function 5 ab, if a. 0 and b. 1, the function represents eponential growth. if a. 0 and 0, b, 1, the function represents eponential deca. In either case, the -intercept is (0, a), the domain is all real numbers, the asmptote is 5 0, and the range is Write D if the function represents eponential deca. Write G if it represents eponential growth. G Problem 5 (5 ) G 5 1 Q7 R D 5 5Q 1 R D Identifing Eponential Growth and Deca Got It? Identif 5 () as an eample of eponential growth or deca. What is the -intercept? 8. Compare 5 () to the function 5 ab. Complete each equation. a 5 b 5 Underline the correct word, epression, or phrase to complete each sentence. 9. The value of b is between 0 and 1 / greater than So, the function represents eponential growth / deca. 11. The -intercept is (0, 0) / (1, 0) / (0, ). 5 (0.5) 187 Lesson 7-1

3 Ke Concept Eponential Growth and Deca You can model eponential growth or deca with this function. Amount after t time periods Rate of growth (r. 0) or deca (r, 0) Initial amount A(t) 5 a(1 1 r) t Number of time periods For growth or deca to be eponential, a quantit changes b a fied percentage each time period. 1. Draw a line from the function in Column A to the situation it models in Column B. Column A Column B B(m) 5 00(1 0.0) m You bu 50 mice. Ever month, the population increases b 15%. P(m) 5 50(1.15) m S(t) 5 8(1.05) t You have 1 lb of dog food. Your dog eats a quarter of it the net da, and each da afterward the dog eats a quarter of what s left. You get a job paing $8 per hour. Ever ear, the salar increases b 5% to keep up with the cost of living. A(d) 5 1(0.75) d Your savings account holds $00. For all accounts under $1000, the bank charges a fee of % per month, drawn from the account. Problem Modeling Eponential Growth Got It? Suppose ou invest $500 in a savings account that pas.5% annual interest. How much will be in the account after five ears? 1. An eponential growth model applies to this problem because the amount in the bank increases / decreases b a fied percentage of.5 % each ear. 1. Complete the model. a 5 the initial amount invested 5 r 5 the rate of growth per ear written as a decimal 5 A(t) 5 a(1 1 r) t Q R t 15. Evaluate the model for t A(t) 5 a(1 1 r) t A 5 500( ) 5 A 5 500(1.05) 5 A 5 $ Chapter 7 188

4 Problem Using Eponential Growth Got It? Suppose ou invest $500 in a savings account that pas.5% annual interest. When will the account contain $650? 16. Complete the model. a 5 r the initial amount invested A(t) 5 a(1 1 r) t Q 1.05 R t 17. Use the TABLE feature on our calculator to complete the table. Use for t and for A(t). 18. The account will contain $650 after 8 ears Lesson Check Do ou UNDERSTAND? Vocabular Eplain how ou can tell whether 5 ab represents eponential growth or eponential deca () represents eponential growth. The value of b in this function is Q 1 R 1 represents eponential deca. The value of b in this function is. 1. Now answer the question. Answers ma var. Sample: If b S 1, the function represents eponential growth. If 0 R b R 1, the function represents eponential deca. Math Success Check off the vocabular words that ou understand. eponential function eponential growth eponential deca Rate how well ou can model eponential growth and deca. Need to review Now I get it! 189 Lesson 7-1

5 7- Properties of Eponential Functions Vocabular Review The figure in blue is the image of the figure in black. Identif each transformation as a translation, reflection, or dilation O translation dilation reflection Vocabular Builder eponential growth (noun) ek spoh NEN shul grohth Opposite: eponential deca Definition: For the function 5 ab, if a. 0 and b. 1, the function represents eponential growth. Eample: A savings account earns percent interest annuall. The amount in the account grows eponentiall. The equation 5 0.5(9) represents eponential growth. Use Your Vocabular. Circle the functions that represent eponential growth. 5.5(0.75) 55(1.0) 5 0.8(.7) 5 6(9.) 5. Cross out the functions that do NOT represent eponential growth. 5 (0.5) 5 6( 9.) 5() 5 (7) Chapter 7 190

6 Problem 1 Graphing 5 ab Got It? How does the graph of 50.5? 5 compare to the graph of the parent function? The graphs of 5 5 (black) and 50.5? 5 (blue) are shown at the right. Underline the correct words or value to complete each sentence. 6. Graphing 50.5? 5 involves a reflection / translation of the parent function. 7. The domain of 50.5? 5 is / is not the same as the domain of the parent function. 8. The range of 50.5? 5 is / is not the same as the range of the parent function. 9. The -intercept of 50.5? 5 is (0, 1) / (0, 0.5). 10. Multipling 5 b 0.5 stretches / compresses the vertical scale. Problem Translating the Parent Function 5 b Got It? How does the graph of 5 (1) compare to the graph of the parent function? 11. The graph of 5 (1) is a horizontal / vertical translation of 5 two units to the left / right. 1. The graph at the right shows 5. Sketch the graph of 5 (1) on the same set of aes. Parent Function Stretch A u au. 1B Compression (Shrink) A0, u au, 1B Reflection (a, 0) in -ais Concept Summar Families of Eponential Functions Translations (horizontal b h; vertical b k) 5 b 5 ab 5 b (h) 1 k 6 O All transformations combined 5 ab (h) 1 k 191 Lesson 7-

7 1. Each blue figure is a transformation of each black figure. Label the transformation as a vertical stretch, vertical compression, reflection, or translation. 1O O 1 stretch reflection translation translation or translation or compression or compression Problem Using an Eponential Model Got It? The initial temperature of a cup of coffee is 0 F. An eponential model for the temperature of the coffee after minutes is 5 1.5? How long does it take for the coffee to reach a temperature of 100 F? 1. Enter the function into our graphing calculator. Then complete the table It takes about 1.9 minutes to cool to 100 F. Ke Concept Continuousl Compounded Interest The formula for continuousl compounded interest uses the number e: amount in account at time t A(t) P e rt Principal Problem 5 Continuousl Compounded Interest interest rate (annual) time in ears Got It? Suppose ou won a contest at the start of 9th grade that deposited $000 in an account that pas 5% annual interest compounded continuousl. You start college ears later, and spend ears in college. About how much will be in the account after ears of college? Chapter 7 19

8 16. The mone is earning interest over 8 ears. 17. Use the justifications at the right to solve the problem. A 5 P? e r?t 5 000? e Q 0.05 RQ 8 R Substitute values for P, r, and t. < $ 75 Use a calculator. Round to the nearest dollar. Lesson Check Do ou UNDERSTAND? Reasoning Is investing $000 in an account that pas 5% annual interest compounded continuousl the same as investing $1000 at % and $1000 at 6%, each compounded continuousl? Eplain. 18. Cross out the equation(s) that ou could NOT use to solve this problem. A 5 000? e (0.05)t A ? e (0.05)t A ? e (0.0)t A Ae (0.05)t 1 e (0.06)t B A ? e (0.06)t 19. Use a calculator to complete the table. Round to the nearest dollar. t 000 e 0.5t 1000 e 0.t 1000 e 0.6t 1000 e 0.t 1000 e 0.6t 1 $ 10 $ 101 $ 106 $ 10 $ $ 118 $ 1197 $ 5 5 $ Now answer the question. $ 11 $ 150 $ 571 No. One account at 5% earns less than two accounts at % and 6%. Math Success Check off the vocabular words that ou understand. natural base eponential function Rate how well ou can graph eponential functions with base e. Need to review Eplanations ma var. Sample: continuousl compounded interest Now I get it! 19 Lesson 7-

9 7- Logarithmic Functions as Inverses Vocabular Review 1. Circle the base in each power. 5 5 ( ) Vocabular Builder logarithm (noun) LAWG uh rith um Related Words: base, power, eponent Definition: The logarithm base b of a positive number is the eponent to which base b must be raised to get. Eamples: Since 5 16, the logarithm base of 16 is. Similarl, since 5 6, the logarithm base of 6 is. Math Usage: A logarithm is written as log b and is read log base b of. Use Your Vocabular Write each logarithm in words. The first one is done for ou. Sample: log 16 log base of 16. log. log. log 5. log log base of log base of log base of log base of Ke Concept Logarithm A logarithm base b of a positive number satisfies the following definition. log b 5 if and onl if b 5 Chapter 7 19

10 6. Draw a line from each logarithm equation in Column A to its eponential equation in Column B. Column A Column B log log 9 5 b 5 log log b Problem 1 Writing Eponential Equations in Logarithmic Form Got It? What is the logarithmic form of the equation 6 5 6? 7. Use the definition of logarithm: If 5 b, then log b 5. Since 6 5 6, log The logarithmic form of the equation is log Problem Evaluating a Logarithm Got It? What is the value of log 5 15? 9. Use the justifications at the right to complete the equations. 5 log 5 15 Write the logarithm Use the definition of a logarithm to write an eponential equation Write 15 as a power of 5. 5 Since the bases are the same, the eponents must be equal. log Write the value of log A common logarithm is a logarithm with base 10. You can write a common logarithm log 10 simpl as log, without showing the 10. Problem Using a Logarithm Scale Got It? In 1995, an earthquake in Meico registered 8.0 on the Richter scale. In 001, an earthquake of magnitude 6.8 shook Washington state. How man times more intense was the 1995 earthquake than the 001 earthquake? Use the formula log I 1 I 5 M 1 M to compare the intensit levels of earthquakes, where I is the intensit level and M is the magnitude on the Richter scale. 10. Circle the equation that models the problem. log I 1 I log I 1 I log I 1 I Lesson 7-

11 11. Simplif the equation ou circled in Eercise 10. log I 1 I Circle the common logarithm that corresponds to the simplified equation. I 1 I I 1 I I 1 I Use a calculator. The 1995 earthquake was about times more intense than the 001 earthquake. Problem Graphing a Logarithmic Function Got It? What is the graph of 5 log? Describe the domain, range, -intercept, and asmptotes. 1. The graph of 5 is at the right. Circle the graph of its inverse 5 log O 8 16 O O Draw a line from each item in Column A to a corresponding item in Column B. Column A Column B Domain of 5 log 5 0 Range of 5 log. 0 -intercept of graph of 5 log. 0 Vertical asmptote(s) of 5 log none all real numbers Concept Summar Families of Logarithmic Functions Parent function 5 log b, b. 0, b 1 Stretch (u a u. 1) Compression (shrink) (0, u a u, 1) 5 a log b Reflection (a, 0) in -ais Translations (horzontal b h; vertical b k) 5 log b ( h) 1 k All transformations together 5 a log b ( h) 1 k Chapter 7 196

12 16. Circle each equation that is a compression of the parent function. Underline each equation that is a reflection of the parent function. 5 log 5 0. log 8 5 log ( 5) log 10 Problem 5 Translating log b Got It? How does the graph of 5 log b ( ) 1 compare to the graph of the parent function? 17. The graph at the right shows the parent function, 5 log. Underline the correct word or value to complete each sentence. To graph 5 log ( ) 1, translate the graph of the parent function / / units left / right and / / units up / down. The domain becomes. /. The range does / does not change. 8 log O Lesson Check Do ou UNDERSTAND? Vocabular Determine whether each logarithm is a common logarithm. log log 6 log log Circle the base of each logarithm above. Complete. 19. Common logarithms have a base of Logarithms without a base are assumed to have a base of Write Y for es or N for no to indicate whether each logarithm is a common logarithm. N log Y log 6 Y log N log 5 5 Math Success Check off the vocabular words that ou understand. logarithm logarithmic function common logarithm logarithmic scale Rate how well ou can use and graph logarithms. Need to review Now I get it! 197 Lesson 7-

13 7- Properties of Logarithms Vocabular Review 1. Circle the logarithms below. log 6 9 log ( 1) a? b (11). Circle the base of each logarithm log log 7 9 log log 5 15 Write T for true or F for false. F T F. If log b 5, then b 5.. If log 5 w, then 10 w If 5, then log 5. Vocabular Builder formula (noun) FAWRM oo luh Other Word Forms: formulate (verb), formulaic (adjective) Definition: A mathematical formula is an equation that ou can use to solve a particular kind of problem. Eample: You can use the quadratic formula to solve quadratic equations. You can use the distance formula to find the distance between two points. Use Your Vocabular 6. Circle the slope formula. A 5 1 (b 1 1 b )h A 5 Pe r t P 5 l 1 w m Chapter 7 198

14 Since 5 log a if and onl if 5 a, logarithms and eponents have corresponding properties. Properties Properties of Logarithms For an positive numbers m, n, and b where b 1, the following properties appl. Product Propert log b mn 5 log b m 1 log b n Quotient Propert log b m n 5 log b m log b n Power Propert log b m n 5 n log b m Use the properties of logarithms to complete each equation. 7. log 0 5 log (5? ) 5 log 5 1 log 8. log 5 ( 9 ) 5 9 log 5 9. log log 5 log 5 log log (5 ) 5 log 5 1 log 5 log 5 1 log Problem 1 Simplif ing Logarithms Got It? What is log 6 log 9 written as a single logarithm? If possible, simplif the single logarithm. 11. Circle the propert ou can use to rewrite log 6. Product Propert Quotient Propert Power Propert 1. Use the propert ou circled above to rewrite the first term of log 6 log 9. log 6 log 9 5 log 6 log 9 1. Circle the propert ou can use to combine the last two terms in Eercise 1. Product Propert Quotient Propert Power Propert 1. Use the propert ou circled above to combine the two terms. 6 log 6 log 9 5 log 5 log Use the definition of logarithm to simplif the epression. log 5 1 Problem Epanding Logarithms Got It? What is log 50 7 epanded? Simplif our answer, if possible. 199 Lesson 7-

15 16. Follow the steps to epand the logarithm. log log 50 log 7 5 log 1 log 15 log 7 Use the Quotient Propert of Logarithms. Use the Product Propert of Logarithms. 5 log 1 log 5 log 7 Write 15 as a power of 5. 5 log 1 log 5 log 7 Use the Power Propert of Logarithms. Properties Change of Base Formula For an positive numbers m, b, and c, with b u 1 and c u 1, log b m 5 log c m log c b. Use the Change of Base Formula to complete each equation. log log log log log log 100 log log 5 log 5 0. Reasoning The base implied in Eercise 17 is / 5 / 10 / 100. Problem Using the Change of Base Formula Got It? What is the value of log 8? 1. Circle the least common factor of 8 and. 8. Complete each equation. Since 5 5, log 5 5. Since 5 8, log log 8 5 log log Got It? What is the value of log 18?. Circle the calculator-read form of log 18. log log 18 5 log 9 log log 9 log 5. The value of log 18 is approimatel.085. Problem 8 5 Using a Logarithmic Scale log 18 log Got It? Chemistr The ph of a substance equals log fh 1 g, where fh 1 g is the concentration of hdrogen ions. Suppose the hdrogen ion concentration for Substance A is twice that for Substance B. Which substance has a greater ph level? What is the greater ph level minus the lesser ph level? Eplain. Chapter 7 00

16 6. If fh 1 bg is the concentration of hdrogen ions for Substance B, circle the ph of Substance B. log fh 1 b g log fh1 b g log 1 fh1 b g log fh1 b g 7. Circle the epression for the concentration of hdrogen ions for Substance A. fh 1 b g? fh1 b g fh 1 b g log fh 1 b g 8. Circle the epression for the ph of Substance A. log f? fh 1 b gg log ffh1 b gg 1 log fh1 b g log fh1 b g 9. Use the Product Propert of Logarithms to epand the epression ou circled above. log f? fh 1 b gg 5 Qlog 1 log fh 1 b gr log fh 1 b g (ph of Substance A) (ph of Substance B) 0. Circle the substance with the greater ph level. Substance A Substance B 1. What is the difference between the ph levels? Eplain how ou know. log, or Eplanations ma var. Sample: The log of twice a quantit is log more than log. Lesson Check Math Success Need to review Do ou UNDERSTAND? Reasoning If log 5 5, what is the value of 1?. Underline the correct epression to complete each sentence. If log 5 5, then > Since > 1, I know > Check off the vocabular words that ou understand. logarithm Change of Base Formula Rate how well ou can use the properties of logarithms Now I get it! 01 Lesson 7-

17 7-5 Eponential and Logarithmic Equations Vocabular Review 1. Underline the base and circle the eponent in each epression. 7 ( 1 ) 1 9 (17). Multiple Choice Which epression is read five to the third power? Vocabular Builder isolate (verb) EYE soh lat Other Word Forms: isolation (noun), isolated (adjective) Definition: When ou isolate something, ou get it b itself. Math Usage: To isolate a variable in an equation means to get the variable alone on one side of the equal sign. Use Your Vocabular. Draw a line from each equation in Column A to the operation(s) needed to isolate the variable in Column B. Column A b 5 15 Column B z Divide both sides b. Multipl both sides b. Then add 5 to both sides. 1 (p 5) 5 Subtract 9 from each side, then divide both sides b. Problem 1 Solving an Eponential Equation Common Base Got It? What is the solution of ?. Circle the common base that ou can use to rewrite 7 and Chapter 7 0

18 5. Rewrite 7 and 81 using the common base. a b 5 6. If two numbers with the same base are equal, their eponents are equal / not equal. 7. Solve for. A B Problem Solving an Eponential Equation Different Bases Got It? What is the solution of ? 8. The equation has been solved below. Write a justification for each step log 5 5 log 10 log 5 5 log 10 5 log 10 log 5 Write the original equation. Take the logarithm of each side. Power Propert of Logarithms Divide each side b log 5 to isolate. 9. Use a calculator to determine the value of to four decimal places. < 1.51 Problem Solving an Eponential Equation With a Graph or Table Got It? What is the solution of ? 10. Write the two equations ou can graph on our graphing calculator to solve this equation. Y Y Use a graphing calculator to find the point of intersection of the two equations. Write the approimate value of. < Lesson 7-5

19 Problem Modeling With an Eponential Equation Got It? Resource Management Wood is a sustainable, renewable, natural resource when ou manage forests properl. Your lumber compan has 1,00,000 trees. You plan to harvest 5% of the trees each ear. How man ears will it take to harvest half of the trees? 1. You are harvesting a fied number / percentage of trees each ear, so using a linear / eponential growth / eponential deca model is reasonable. 1. Use the phrases from the blue bo to label each part of the formula. amount after n periods number of time periods T(n) a(1 r) n rate of deca initial amount amount after n periods number of time periods initial amount rate of deca 1. Use the information in the problem to write each amount. a 5 1,00,000 r T(n) 5 600, Use our values for a, r, and T(n) to write an equation. 600, ,00,000(1 0.05) n 16. Solve our equation for n, the approimate number of ears it will take to harvest half the original trees. Problem 5 Solving a Logarithmic Equation Got It? What is the solution of log ( ) 5 1? 17. Circle the equivalent eponential form of log ( ) (1) 10 () log( ) Solve the eponential equation for. 600, ,00,000(1 0.05) n 600,000 1,00,000 5 (0.95)n log n log 0.95 log 0.5 log n n N Chapter 7 0

20 Problem 6 Using Logarithmic Properties to Solve an Equation Got It? What is the solution of log 6 log 5? 19. You can use the Product / Quotient / Power Propert of Logarithms to write an equivalent equation with a single logarithm. 0. Write the single logarithm equation equivalent to log 6 log 5. log 5 1. Solve the single logarithm equation for. log Lesson Check Do ou UNDERSTAND? Error Analsis Describe and correct the error made in solving the equation.. If log z 5 log b, does 5? Eplain. Answers ma var. Sample: Onl when the bases are equal (z 5 b) log = log 9 log = log 9 = 9 = 81 will 5.. Complete the equation below.. Correct the error and solve for. log 9 5 log log 5 log 9 5 log log Math Success Check off the vocabular words that ou understand. eponential equation Need to review logarithmic equation Rate how well ou can solve eponential and logarithmic equations. Now I get it! 05 Lesson 7-5

21 7-6 Natural Logarithms Vocabular Review Write T for true or F for false. T F F 1. The function 5 log b, where b. 0 and b 1 is called a logarithmic function.. A logarithmic equation is an equation that contains onl one logarithm.. The logarithm of a power is the difference of the logarithm and the eponent. Vocabular Builder inverse function (noun) IN vurs FUNGK shun Related Words: function, inverse, input, output Definition: To find the inverse function, switch the order of the elements in the ordered pairs of the function. Eample: function, f(): {(1, ), (, )}; inverse function, f 1 (): {(, 1), (, )} Use Your Vocabular. Complete the table of values for the inverse function, f 1 (), of the function f(). 1 f() f 1 () Chapter 7 06

22 Ke Concept Natural Logarithmic Function If 5 e, then 5 log e 5 ln. The natural logarithmic function is the inverse of 5 e, so ou can write it as 5 ln If 5 e 5, then ln If ln b 5 6, then b 5 e. 1 = e = ln Problem 1 Simplifing a Natural Logarithmic Epression Got It? What is ln 7 1 ln 5 written as a single natural logarithm? 7. The epression is simplified below. Write a justification for each step. ln 7 1 ln 5 ln 7 1 ln 5 ln 7 1 ln 5 ln (7? 5) ln 175 Write the original epression. Power Propert of Logarithms Simplif the second term. Product Propert of Logarithms Multipl. Problem Solving a Natural Logarithmic Equation Got It? What are the solutions of ln 5? 8. Complete: If ln 5, then e 5. Got It? What are the solutions of ln ( 1 5) 5? Check our answers. 9. The equation is solved below. Write a justification for each step. ln ( 1 5) 5 ln ( 1 5) 5 e ln ( e ln ( e ln ( (5 e ) Write the original equation. Write in eponential form. Take the square root of each side. Subtract 5 from each side. Divide each side b. ln (0 1 5 < or <.167 Use a calculator. 07 Lesson 7-6

23 10. Check Substitute our values for in ln ( 1 5) 5. Use a calculator. ln Q? R <.0001 ln Q? R <.9976 Got It? What are the solutions of ln 1 ln 5? Check our answers. 11. Circle the propert of logarithms that justifies writing ln 1 ln as ln 6. Power Propert Product Propert Quotient Propert 1. Use the simplified equation to solve for. 1. Check Substitute our value for in ln 1 ln 5. Use a calculator. ln 6 5 ln? ln < 6 5 e e 6 N 1.15 Problem Solving an Eponential Equation Got It? What is the solution of e 5 1? Check our answer. 1. Use the justifications at the right to solve the equation. e ln 1 Write the original equation. Rewrite in logarithmic form. 5 ln 1 1 Add to each side. <.89 Use a calculator. 15. Check Substitute our value for in e 5 1. Use a calculator. e.89 < Got It? What is the solution of e 5 0? Check our answer. 16. Circle the first step in solving the equation. Underline the second step. Divide each side of the equation b. Write in eponential form. 17. Now solve the equation. Use the Power Propert. 18. Check Substitute our value for in e 5 0. Use a calculator. e. < e 5 0 e ln 10 5ln Write in logarithmic form. Chapter 7 08

24 Problem Using Natural Logarithms Got It? Space A spacecraft can attain a stable orbit 00 km above Earth if it reaches a velocit of 7.7 km/s. The formula for a rocket s maimum velocit v in kilometers per second is v t 1 c ln R. The booster rocket fires for t seconds and the velocit of the ehaust is c km/s. The ratio of the mass of the rocket filled with fuel to its mass without fuel is R. Suppose a booster rocket for a spacecraft has a mass ratio of about 15, an ehaust velocit of.1 km/s, and a firing time of 0 s. Can the spacecraft achieve a stable orbit 00 km above Earth? 19. Identif the values for each variable. R 5 15 c 5.1 t Complete the equation for finding the spacecraft s maimum velocit. Then use a calculator to simplif the equation. v ? 0 1 Q.1 ln 15 R km/s < 5.99 km/s 1. The spacecraft will / will not achieve a stable orbit 00 km above Earth. Eplain. Answers ma var. Sample: The velocit of 5. km/s is less than the 7.7 km/s needed to attain a stable orbit. Lesson Check Do ou UNDERSTAND? Reasoning Can ln 5 1 log 10 be written as a single logarithm? Eplain our reasoning.. The ln 5 1 log 10 can be written as ln 5 5 log e 5 1 log 10.. Can ou use the Product Propert of Logarithms to combine the logarithms? Eplain wh or wh not. No; the two logarithms have different bases.. ln 5 1 log 10 can / cannot be written as a single logarithm. Math Success Check off the vocabular words that ou understand. function logarithm natural logarithmic function Rate how well ou can write and solve equations with natural logarithms. Need to review Now I get it! 09 Lesson 7-6