1 Module for Year 10 Secondary School Student Logarithm

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5 Erthquke Intensity Mesurement (The Richter Scle) Dr Chrles Richter showed tht the lrger the energy of n erthquke hs, the lrger mplitude of ground motion t given distnce. The simple model of Richter mgnitude using pproprite units is s follow: M = log 10 A Where A is Amplitude ground motion M is Richter Mgnitude Here some typicl effects of erthqukes in vrious mgnitudes rnge. Mgnitude Effect less thn,5 Generlly not felt, but recorded Often felt, but rrely cuse dmge Slight dmge to well-designed building Destructive Mjor erthquke 8 or greter Severe erthquke, Severe dmge Chrles Richter nd Beno Gutenberg develop model for the reltionship between energy nd mgnitude of resulting erthquke log E = M Where E is energy in Joule M is the Richter mgnitude 4

6 Wht do you expect to lern? This module is designed for you to: 1. Define the logrithmic function f(x) = log x s the inverse of the exponentil function f(x) = x. Understnd nd pply the lw nd properties of the logrithm; nd. Solve simple logrithmic equtions. Motivtion nd definition The ide of logrithms is to reverse the opertion of exponentition tht is rising number to power. For exmple, the third power (or cube) of 4 is, becuse 64 is the product of three fctors of : 4 = = 64 It follows tht the logrithm of 64 with respect to bse 4 is, so log 4 64 =. In contexts to the mgnitude of erthqukes in Richter scle in which opertes on logrithmic bsis so there is 10-fold increse from one unit to the next. An erthquke rted 4 on the Richter scle hs been ssigned reltive intensity of 1 unit. Bsed on this tble, you cn lern bout the comprison of the mgnitude of erthquke. The Reltionship between Richter Scle nd Reltive Intensity Richter Scle (RS) Reltive Intensity Wht cn you sy bout the Reltionship between Richter Scle nd reltive intensity? Definition Look t problem below Find the vlue of n! 10 n = = n =. n = =. n =. b= x x = log b (>0, 1, b> 0) We cn solve the problem bove using logrithm. Wht is the logrithm? From the logrithm definition we cn find the vlue of n 10 n = 100 n = log = log10 n = 5

7 n = n = n =. Some Importnt Result. Since 1 = log 1 = Since = log = b. log 10 is clled the common logrithm nd just write log (without writing bse) c. Nturl logrithm written s Ln is logrithm with bse e (e = ) Let s do Exercises 1. Write in logrithm form.. 15 = 5 b. = c. = 5. Find the vlue. log 16 c. log b. log 4 d. log e. log f. log 10 x The Logrithm The logrithm of number b with respect to bse is the exponent to which hs to be rised to yield b. In other words, the logrithm of b to bse is the solution x of the eqution b = x The logrithm is denoted "x = log (b)" (pronounced s "the logrithm of b to bse " or "the bse- logrithm of b").for the logrithm to be defined, the bse must be positive rel number not equl to 1 nd b must be positive number. Logrithm bse 10 is denoted log b 6

8 Lws of Logrithm Since M = nd N = M. N =. M. N = From the definition b = x x = log b We cn write M. N = (The Lw I) Since M = nd N = =. = Since From the definition b = x x = log b We cn write = (The Lw II) M = M = ( ) From the definition b = x x = log b We cn write M = (The Lw III) Properties of the logrithm log b ) b b) log b log b c log c c) log b log c = log c m n m d) log b log b n b Prove the properties by yourself using definition of logrithm nd the lw of logrithm! 7

9 Exmple: 1. Simplify the following: 1 1. log log log 4 4 b. log6 9 log6 64 log6 16 From the logrithm property: log b c log b log c log log log 4 = log = log log = 1 = 1, then b. log6 64 log6 = log 6 log = log 6... = If log 5 p, then log 75=... (in p form) log 5 75 = (5 ) 5 log 5 log b c log b log c = log 5 5 log 5 = log 5 5 p = + p b. If log 9 8 = n, then log 4 is... log 9 8 = log log 4 = 1 log n = log... =...log... n = log =... 8

10 . The vlue of log 4 nd log respectively re 0,601 nd 0,4771. Find out the vlue of log 6 nd log 48! log 6 = log 4 9 log 48 = log = log 4 + log = log... + log... = log 4 +. log =...log... + log... = 0, ,9541 = = 1,556 =... Logrithmic Equtions Logrithmic equtions re equtions involving logrithmic functions.from previous lesson, you lerned tht exponent ndlogrithmre inverses. Hence, the properties of exponent nd logrithm cnbe used to solve the simple logrithmic eqution Exmple: Solve the missing terms in the following logrithmic equtions. 1. log 6 16 = x Trnsform log 6 16 = x in exponentil form, then solve for x. log 6 16 = x x = y y = log x 6 x = 16 6 x = 6 x =. log 4 (5x + 4) = log 4 (5x 4) log4 (5x 4) log4 64 ( 5x 4) 64 x 1. log(x + 6) = log (x 6) log (x 6) log... ( x 6)... x... 9

11 Logrithmic Appliction for Erthquke Cses 1. How much more intense n erthquke of mgnitude 6.5 s compred to one of mgnitude 4.5? M 1 = log A 1 M = log A 6.5 = log A = log A A 1 = A = So, A 1 = A = 10 = 100 The movement in the ground for erthquke with mgnitude 6.5 is pproximte 100 times greter thn tht of with mgnitude On 11 Jnury 01, Meulboh, Nngroe Aceh Drusslm experienced n erthquke of Richter mgnitude 7.. Clculte the energy ssocited with this erthquke! Exercise Fill in the blnk 1. log 6 log10 log 5 log 4.. If log x 64, then the vlue of x = log 8 log 9 log 7 = log.log 5 8.log. 5 10

12 1 5. If log 0. 5 x, then the vlue of x = logb logc log bc. 7. If 4log4 x 6, then the vlue of x =. 8. log p. 1 p 9. There re log7 p, log q. Trnsform log 6 98 in p nd q form is... x 10. The vlue ofx of logrithmic eqution log is... Explin your nswer for the following question! 11. How much more intense n erthquke of mgnitude 8.4 s compred to the other one of mgnitude 6.8? 1. On 11 Mrch 011, Ishinomki city Jpn experienced n erthquke of Richter mgnitude 9.0. Clculte the energy ssocited with this erthquke! 11

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