LEARNING OUTCOMES AND ASSESSMENT STANDARDS. Converting to exponential form Solve for x: log 2. x = 3

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1 Lesson LOGARITHMS () LEARNING OUTCOMES AND ASSESSMENT STANDARDS Learning outcome 1: Number and number relationships Assessment Standard 1.1. Demonstrates an understanding of the definition of a logarithm and any laws needed to solve real life problems (eg Growth and decay). Overview In this lesson you will: learn how to change the base of a logarithm use the change of base law to simplify expressions and solve equation use your calculator correctly solve real life problems. DVD Lesson Log equations Converting to exponential form Solve for x: log x = Restriction on x is x 0 So: = x x = Dropping the logarithm both sides Solve for x: log (x ) + log (x ) = 1 Restriction on x is x > 0 x > x > 0 x > Thus: x > log (x )(x ) = 1 Change 1 into log log (x )(x ) = log Both sides now have log x x + = Solve by dropping logs x x + = x x + 4 = 0 (x 4)(x 1) = 0 x = 4 or x = 1 x = 4 only since x > 4 Converting to log form: Solve for x: x = 9 LC G1 Log LWB.indb 4 00/09/0 0::9 PM

2 x = _ log 9 log ( def: if ax = b then x = _ log b log a) Use your calculator x = 1, Change of base law log m x = _ log x a log a m Examples 1. Evaluate: lo g 1_ log 9 7 = _ log log _ 1 + _ log 7 log 9 = _ log + _ 1 = _ log 1log + _ = + _ = _ Prime factors. If log x = log y 4, show that y = x log _ log x log y _ log x = _ log 4 Put numbers on one side log y = _ log log _ log y log x = log y = log x log y = log x y = x Use your calculator to evaluate log (Use the log button) Real life applications 1. The radioactive decay of a certain substance is given by the formula m(t) = 00(0,9) t where m is the mass of radioactive substance in grams and t is the number of years. a) What was the initial mass of the radioactive material? Make t = 0 m(0) = 00(0,9) 0 = 00 gms b) By what percentage does the radioactivity decrease by each year? (1 i) 1 = 0,9 i = 0,0 r = % LC G1 Log LWB.indb 00/09/0 0::40 PM

3 c) Determine the mass of the substance after 0 years m(0) = 00(0,9) 0 = 7,7 gms d) After how many years will the substance s mass be less than 1 gram? 00(0,9) n < 1 (0,9) n < _ 1 00 n > log _ 1 00 log 0,9 since log 0,9 and log _ 1 are both negative 00 n > 74, n = 7 Remember: the log of any number between 1 and 0 is always negative. When we have an inequality and we divide by a negative, the sign must swap around. The Richter Scale was developed in 19 by American scientist Charles F Richter (1900 to 19). It is used to measure the magnitude of an earthquake by taking the logarithm of the amplitude of waves recorded by a seismograph. The magnitude of an earthquake is defined by M = log _ I s, where I is the intensity of the earthquake (measured by the amplitude of a seismograph reading taken 100 km from the epicentre of the earthquake) and S is the intensity of a standard earthquake (the amplitude of which is 1 micron = 10 4 cm). a) Calculate the magnitude of a standard earthquake. b) If the intensity of the Ceres earthquake was x micron, express log S in terms of x and the magnitude of the Ceres earthquake. c) If the intensity of the Sumatra earthquake was y micron, express log S in terms of y and the magnitude of the Sumatra earthquake. d) Determine the value of _ y x. Explain what this value means. Activity 1. Calculate correctly to three decimal places:. Solve for x: 14 x = log,4. Solve: x = 4. Solve for x: x 11. x + 4 = 0 Activity 1. Solve for x: a) x + 1 = x b) 7 x = 0, c) 7 x + 7 x + = 0. If x = prove (no calculator) that x = 1 +. LC G1 Log LWB.indb 00/09/0 0::41 PM

4 ANSWERS AND ASSESSMENT Lesson 7. log 0 _ = log = 1. log log 1 = log_ ( ) = log = log = log = 4. log + log log = log (_ ) = log 100 = log 10 = log 10 =. 1_ log log + _ 1 log 4 = _ 1 log () log + _ 1 log () {prime factorise} = log log + log = log + log = log (.) {law 1} = log 10 = 1 Activity 1. log _ 1 = log _ 1 = log 1 = 1 log. 1 _ log = _ log log = _ log 4 4log = _ 4. (1 9) = = log 4. 1 _ = _ 4 = log 4 + log. log log + log log log log + log log log = _ 7 log 4 log = _ 7 4. log 9 (log ) = log 9(log ) = log 9 ( log ) = log 9 = _ log 9 = _ = _ 1 log log 7. log 4log 10. ()(4) = 1 (log log ) (log log ) = _ 1. log 4 ( 1 _ 1 1 _ 1 _ 1 ) = log 4 1 _ 4 = log = log + log 7 7 = log + log 7 7 = + = 1. ( 1_ 4 ) + 1_ = _ 4 + _ 1 = _ log ( _ ) + log ( _ ) + log ( 1_ ) log 10 = log ( 9 ) = log 10 = log = Lesson a) Calculate the magnitude of a standard earthquake. M = log _ I S I = 10 4 S = 10 4 M = log _ M = log 1 m = 0 b) If the intensity of the Ceres earthquake was x micron, express log S in terms of x and the magnitude of the Ceres earthquake. M = _ x S, = log x log s log S =, log x c) If the intensity of the Sumatra earthquake was y micron, express log S in terms of y and the magnitude of the Sumatra earthquake. 9,1 = log _ y S 9,1 = log y log S log S = log y 9,1 d) Determine the value of _ y x. Explain what this value means. log x, = log y 9,1, = log y log x, = log _ y x 10, = _ y _ y x x = 1 Intensity of the Sumatra earthquake 1 times that of the Ceres earthquake. Activity 4 1. = _ log = 1,9 [keys log log = ] 14 x = log,4. 14 x = (log,4) log (log,4) x log 14 = 0.. x = log x = _ log = 1,4 4. let x = k k 11k + 4 = 0 (k )(k ) = 0 x = or x = x = _ log or x = x = 1, or x = 7 LC G1 Log LWB.indb 7 00/09/0 0::4 PM

5 Activity 1. a) x+1 = x x = x (_ ) x = _ 1 1 x = log ( _ 1 log (_ 1 ) ) x = 7,1 b) 7 x = 0, log 0, x = _ log 7 = 0, c) 7 x + 7 x 7 = 0 let 7 x = k 0k = 0. x = _ log k = 7 x = x = _ log log 7 x = 0, + log = 1 + TIPS FOR THE TEACHER Lesson 7 Logs are easier that they were in the previous syllabus. A lot of emphasis must be placed on using logs to solve equations and real-life applications. Lesson Logs are easier than they were in the previous syllabus. A lot of emphasis must be placed on using logs to solve equations and real life applications. LC G1 Log LWB.indb 00/09/0 0::4 PM