Raffles Institution 2024 Y3 SUPP WS Exponential and Logarithmic Eqns and Fn

Page 1 of 4 RAFFLES INSTITUTION MATHEMATICS DEPARTMENT 2024 YEAR 3 RP MATHEMATICS TOPIC 9:EXPONENTIAL AND LOGARITHMIC EQUATIONS & FUNCTIONS (MATHS 1 & MATHS 2) SUPPLEMENTARY WORKSHEET Name: Class: Sec 3 ( ) Date: 1 2022/Y3RP/M2/T2/Q1 Solve the equation ( ) ( )6 6 0.2 1log 2 3 4 log 4 log 25xx+ − − = . [3] [Ans: 76 15 ] 2 2022/Y3RP/M2/T2/Q2 Solve the equation 1 11 2 3 2 8 0 xxe e e ++ − − = , giving your answer(s) correct to 3 significant figures. [4] [Ans:1.39] 3 2022/Y3RP/M2/T2/Q3 Given that 2loga xm= and log a yn= , express 4logx y in terms of m and n . [4] [Ans: n m ] 4 2022/Y3RP/M2/T2/Q4 (i) Sketch the graph of ( )ln 2 3yx=+ , indicating clearly the asymptote and any intersection(s) with the axes. [2] (ii) Insert on your sketch the additional straight line graph required to obtain a graphical solution of the equation 3 23 xe x= + . [2] [Ans: ln 3yx=− + ] 5 2022/Y3RP/M2/T2/Q5 Carbon dating is a method used to estimate the age of archaeological artefacts. The age, W years old, of an artefact with a carbon ratio of N is given by ln 57000.693 NW =−  . (i) Find the age of an artefact, correct to the nearest 1000 years, with a carbon ratio of 0.48. [1] (ii) Express N in the form kWNe= , where k is a constant correct to 3 significant figures. [2] (iii) Hence, sketch the graph of N against W . [2] [Ans: (i) 6000 years old (ii) 0.000122WNe −= ] 6 2021/Y3RP/M2/T2/Q1 Solve 2 1lg(ln ) 2x = . [2] [Ans: 4.86 ] 7 2021/Y3RP/M2/T2/Q2 Solve ( )( ) 2 2 13 2 5 15x x x +−= . [4]

Page 2 of 4 [Ans: 0.683− ] 8 2021/Y3RP/M2/T2/Q3 Sketch the graph of ( )ln 5 2yx=− . Insert in your sketch the additional straight line required to obtain a graphical solution to the equation ( ) 33 52xex+ =− . [4] [Ans: 1 13yx=+ ] 9 2021/Y3RP/M2/T2/Q4 Solve the simultaneous equations 27 39 x y= , 3 3 3log 1 log log ( 4)x y x− + = + . [5] [Ans: x = 3, y = 7] 10 2021/Y3RP/M2/T2/Q5 Given that 8log xp= and 8log yq= , find (i) m in terms of p if 64 2 mx= , [2] (ii) log 4x y in terms of p and q. [3] [Ans: (i) 63mp=+ (ii) 46 3 q p + ] 11 2020/Y3RP/M2/T2/Q1 (i) State the equation of the asymptote of the graph of ( )ln 2 3yx=− . [1] (ii) To solve the equation 1 32xex+ += using the graph of ( )ln 2 3yx=− , a straight line graph needs to be inserted. Determine the equation of this straight line. [2] [Ans: (i) 2 3x= (ii) 11 22yx=+ ] 12 2020/Y3RP/M2/T2/Q4 Without using a calculator, solve the equation ( ) ( ) 2 16 27 22log 2 1 6log 9 log 5 3xx− + = − . [5] [Ans: 3 5x= ] 13 2020/Y3RP/M2/T2/Q5 (i) Solve the equation 3 61(2 ) 15 2x x++= . [3] (ii) Hence, without using a calculator, express 8log 72 in terms of x where x is the solution of the equation in (i). [2] [Ans: (i) 0.528 (ii) 2x + 1] 14 2019/Y3RP/M2/T2/Q1 Given that 27log px= , find 3x in terms of p. [2] [Ans: 33x p= ] 15 2019/Y3RP/M2/T2/Q5 Solve the following equations, leaving your answer correct to 3 significant figures where necessary. (a) ( )ln 8 3 ln3 ln 2x x− − = [4] (b) ( )33log 3 6log 3 log xxxx ++ − = [5] [Ans: (a) 0.893 (b) 24]

Page 3 of 4 16 2019/Y3RP/M2/T2/Q6 In a laboratory experiment to determine conditions suitable for amoeba, a researcher studied a sample which initially contained 3 500 000 amoebae. The observed size of the amoeba population, N, is given by 0 atN N e= where 0N and a are constants and t is the time in days since the start of the experiment. Due to strongly unfavourable conditions, it was observed that 30% of the population died each day. (i) Determine the values of 0N and a. [3] (ii) Calculate the number of days that will pass before the population reduces to 2000, giving your answer to the nearest day. [2] [Ans: (i) 0.357− (ii) 21 days] 17 2015/Y3RP/T2/Q5 Solve the equation: (a) 135x+ = , [2] (b) ( ) ( )5 5 5log 1 log 2 2log 6xx− + − = , [3] (c) 11 30xxe e e+−−= . [3] [Ans: (a) 0.465 (b) 4 (c) 1.79] 18 2013/Y3RP/T2/Q1 Solve 212 2 8xx +−= . [4] [Ans: x = 2] 19 2012/Y3RP/T2/Q3 Solve the equation 4 3 2 752xx−+ = , giving your answer correct to three significant figures. [3] [Ans: 1.92] 20 2012/Y3RP/T2/Q4 Solve the equation ( ) ( )5 5log 3 11 log 1 1xx+ = − + . [4] [Ans: 2 5− ] 21 2012/Y3RP/T2/Q5 Sketch the graph of ( ) 3ln 2 3 , 2y x x= −  . Insert in your sketch the additional graph required to obtain a graphical solution to the equation 223 xxe −=+ . [4] [Ans: y = 2 – x] 22 2012/Y4RP/T1/Q1 Given that 3log xa= and 3log yb= , express ( ) 4 3 3log x y in terms of a and b. [3] [Ans: 144 2ab+− ] 23 2012/Y4RP/T1/Q2 Solve the following equations (i) 34xxee −+= , leaving your answers in logarithmic form where necessary, [4] (ii) 63log log 6 2 yy−= , leaving your answers to 3 significant figures where necessary. [5] [Ans: (i) −ln3 or 0 (ii) 0.550 or 6] 24 2012/Y4RP/T1/Q3 Sketch the graph of ( )ln 3yx=+ for x > –3. Determine the equation of the additional straight line which would need to be drawn in order to obtain a graphical solution of the equation ( ) 42 3xex+ =+ . [4]

Page 4 of 4 [Ans: 11 42yx=+ ] 25 2009/Y3RP/T4/Q1 Given that lg xa= and lg yb= , express 3 5 2lg 100 x y    in terms of a and b. [3] [Ans: ( )1 5 2 23 ab−− ] 26 2009/Y3RP/T4/Q2 Solve the equation (a) ( )( ) 1 2 22 5 10x x x−+ = , giving your answer to 3 significant figures, [4] (b) 2 3log 2 12log 3 yy −= . [4] [Ans: (a) 1.10 (b) 127 or 9 ] 27 2009/Y3RP/T4/Q3 (i) Given that 5log xa= and 25log yb= , express 2xy and 2x y as powers of 5. [3] (ii) Further given that 2 625xy = and 2 1 25 x y = , find the values of a and b. [3] [Ans: (i) 4 2 25 , 5a b a b+− (ii) a = 0, b = 1] 28 2009/Y3RP/T4/Q4 Sketch the graph of ( )ln 2yx=− for x > 2. On the same axes, insert the graph of the additional straight line which would need to be drawn in order to obtain a graphical solution of the equation ( ) 23 2xe x e −= . [5] [Ans: y = 3 – 2x] 29 2008/Y3RP/T3/Q2 Solve 1 15 21 32 0 5 x x+ − + = . [4] [Ans: −0.317] 30 2008/Y3RP/T3/Q3 Given that 12log log 4x x yy+= , where x, y > 0 and ,1xy  , express y in terms of x in its simplest form. [3] [Ans: y = x4] 31 2008/Y3RP/T3/Q5 Solve the simultaneous equations ( )22 813, 9 log 3 2 3 log . x y yx = − − = [5] [Ans: x = 2, y = 6] 32 2008/Y3RP/T3/Q6 Solve ( ) ( ) 1ln 2 2 ln 2 1 2 ln 2 2ln 2xx x++ + − = + . [5] [Ans: x = −0.585] 33 2007/Y3RP/T3/Q3 Solve the equation 16xxee −−= , leaving your answers in the form lnab , where a and b are integers. [3] [Ans: ln3]