F-IF.8. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.

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1 Unit 5: Lesson4 Aim: How do we use the properties of logarithms to solve word problem? F-IF.8. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. Lesson Note: By the end of the lesson, students will be able to use the properties of log to solve word problems that lead to equations that are otherwise difficult or impossible to solve. The DoNow is a review of the topic learned in previous lesson. The main lesson focuses on using the properties of log to break down a complex equation so that it can be solved. DoNow: Solve for x: 2log ' x log ' (x + 6) =. / 0 log 1 =. ' 023 / 0 1 = = x / 3x 18 = 0 (x 6)(x + 3) = 0 x = 6, x = 3 rej {6} Homework: Review previous homework New homework is given on separate sheet Mini-Lesson: 1) An archaeologist can determine the approximate age of certain ancient specimens by determine the amount of carbon-14 by using the formula, A = A C 2FGHI, DE where A is the amount of carbon-14 a specimen contains, A C is the original amount of carbon-14, t is time, in years, and 5760 is the half-life of carbon-14. A specimen originally contained 120 milligrams of carbon-14 now contains 100 milligrams of this substance. What is the age of the specimen, to the nearest hundred years? 100 = 120 K2 DE FGHIL.CC = 2 DE FGHI./C ln.cc DE = ln 2FGHI./C ln.cc = NO ln 2./C PQ3C RS 5II 51I = NO RS / PQ3C t = 5760 V RS5II 51I RS / W t 1515 = 1500 Scaffolding: For struggling students, it may be easier to find the values of.1823 and ln so they are not ln.cc./c burdened with the complicate-looking expressions of ln./c and ln 2 in the calculation process. It may also be helpful to review the properties of log.cc

2 Step 1: from the problem, identify A = A C = t = Step 2: substitute values into equation/formula Step 3: divide 120 on both sides to help isolate t Step 4: take log/ln on both sides Step 5: move the exponent of NO to the front of log/ln PQ3C Step 6: solve for t Ask: 1) Why can t we divide by 2 on both sides after step 2? 2) Instead of using ln in step 3, what is another option? How will that affect the answer? How do we know? 2) Drew s parents invested $1,500 in an account such that the value of the investment doubles every 7 years. The value of the investment, V, is determined by the equation V = V C (2) E G, where V C represents the initial investment and t represents the number of years since the money was deposited. How many years, to the nearest tenth of a year, will it take the value of the investment to reach $1,000,000? 1,000,000 = 1500(2) E G.,CCC,CCC.PCC ln.,ccc,ccc.pcc = 2 E G = O Q ln 2 t = 7 V RS5,III,III 5FII W RS / t years 3) Describe the steps that you took to solve the equation in #2. Step 1: divide 1500 on both sides to isolate 2 E G Step 2: take the log (ln ) on both sides of the equation. Step 3: move the exponent O to the front of log 2 Q Step 4: multiply 7 on both sides Step 5: divide by log 2 on both sides to isolate t Informative Assessment: In solving the equation 4 = 2(1.3) O, Jerry first divided both sides by 2 to get 2 = Which is NOT a valid next step: a) log 2 = log b) ln 2 = ln c) 1.3 =

3 4) Since January 1980, the population of the city of Brownville has grown according to the mathematical model y = 720,500(1.022) 0, where x is the number of year since January a) Explain what the numbers 720,500 and represent in this model. 720,500 is the population of Brownville in January means the population is growing at 22% annually ( ) 100 = 22% b) If this trend continues, use this model to predict the year during which the population of Brownville will reach 1,548,800. 1,548,800 = 720,500(1.022) 0 x = ) The current population of Little Pond, New York, is 20,000. The population is decreasing, as represented by the formula P = A(1.3) NC./jkO, where P = final population, t = time, in years, and A =initial population. a) What will the population be 3 years from now? Round your answer to the nearest hundred people. P = 20,000(1.3) NC./jk(j) P = b) To the nearest tenth of a year, how many years will it take for the population to reach half the present population? 10,000 = 20,000(1.3) NC./jkO.C,CCC /C,CCC = 1.3NC./jkO ln.c,ccc /C,CCC t = = t ln 1.3 RS5I,III 1I,III NC./jk RS..j t Summary: How do we use the properties of log to solve word problems?

4 Exit Ticket: The number of houses in Central Village, New York, grows every year according to the function H(t) = 540(1.039) O, where Hrepresents the number of houses, and t represents the number of years since January a) explain, with proper units, the meaning of 540 and in the context of the problem b) A civil engineering firm has suggested that a new, larger well must be built by the village to supply its water when the number of houses exceeds 1,000. During which year will this first happen?

5 MRN22 Word Problems Using Log Name ) An archaeologist can determine the approximate age of certain ancient specimens by determine the -t amount of carbon-14 by using the formula, 5760 A= A 2 0, where A is the amount of carbon-14 a specimen contains, A is the original amount of carbon-14, t is time, in years, and 5760 is the half-life of carbon A specimen originally contained 120 milligrams of carbon-14 now contains 100 milligrams of this substance. What is the age of the specimen, to the nearest hundred years? 2) Drew s parents invested $1,500 in an account such that the value of the investment doubles every 7 years. The value of the investment, V, is determined by the equation V = V C (2) G, E where V C represents the initial investment and t represents the number of years since the money was deposited. How many years, to the nearest tenth of a year, will it take the value of the investment to reach $1,000,000?

6 4) Since January 1980, the population of the city of Brownville has grown according to the mathematical model y = 720,500(1.022) x, where x is the number of year since January a) Explain what the numbers 720,500 and represent in this model. b) If this trend continues, use this model to predict the year during which the population of Brownville will reach 1,548,800. 5) The current population of Little Pond, New York, is 20,000. The population is decreasing, as represented t by the formula P = A(1.3), where P = final population, t = time, in years, and A = initial population. a) What will the population be 3 years from now? Round your answer to the nearest hundred people. b) To the nearest tenth of a year, how many years will it take for the population to reach half the present population?

7 MRN11 HW 23 Name ) Solve for x: log (0N/) (2x) + log (0N/) (x 3.5) = 2 2) The number of houses in Central Village, New York, grows every year according to the function Ht () = 540(1.039) t, where H represents the number of houses, and t represents the number of years since January a) explain, with proper units, the meaning of 540 and in the context of the problem b) A civil engineering firm has suggested that a new, larger well must be built by the village to supply its water when the number of houses exceeds 1,000. During which year will this first happen?

8 3) Solve for x in simplest a + bi form: 2x / 6x + 5 = 0 4) Solve for x in simplest radical form: x / 6x = 4 August 2016: # 1, 3, 5,

9 MRN22 HW 24 Name ) Solve for x to the nearest hundredth. (hint: use quadratic formula instead of factoring) log / (x / 7x + 12) log / (2x 10) = 3-0.1m 2) After an oven is turned on, its temperature, T, is represented by the equation T = (3.2), where m represents the number of minutes after the oven is turned on and T represents the temperature of the oven, in degree Fahrenheit. a) To the nearest degree Fahrenheit, what is the temperature of the oven after it is turned on for 10 minutes? b)to the nearest minutes, how many minutes does it take for the oven s temperature to reach 300 F? June 2016: # 2, 33, 36