SAJC Check your Understanding Exponential & Logarithmic Functions teacher

Check your Understanding (Exponential & Logarithmic Functions) Section 1: Exponential Functions 1. By letting xye= , find the values of x for which 21xxee −−= . [4] 21xxee −−= 2 1y y − = 2 20yy − − = ( )( )2 1 0yy − + = 2 or 1y = − 2 or 1xe = − (reject 1− since 0xe  ) ln 2x= 2. By means of the substitution 212 xu += , or otherwise, solve the equation: 21 21 13 222 x x + + −= [3] ( )( ) 2 2 13 2 2 2 3 2 3 2 0 2 1 2 0 uu uu uu uu −= −= + − = − + = 2 1 1 1 22 122 2 2 1 1 1 x u or u x x +− = =− == + =− =−

3. By using the substitution 3xy= , or otherwise, solve the equation ( ) 11 22 3 9 53 x x +− −= . [5] ( )32 9 3 533 x x  −= Let 3xy= . 32 9 53 yy  −= 29 53 6 0yy+ − = ( )( )9 1 6 0yy− + = 1 9y= or 6y=− So, 13 9 x = or 36x =− (reject) 2x=− 4. Find the exact answer(s) for the equation 3ex = 2e–x – 1. [4] Given 3 2 1xxee −=− , Let xye= , 231y y=− 23 2 0yy+ − = (3 2)( 1) 0yy− + = 2 or 13y=− . So 2 or 1 [N.A.]3 xxee= =− 2ln .3x=

5. Show that 31 13 27 9 y x− =  can be expressed as 3 2 4xy+= . [2] 31 13 27 9 y x− =  3 1 3 23 3 (3 )xy−−= 3 1 3 23 3 (3 )xy−−= 3 1 3 233xy−−= 3 1 3 2xy− = − 3 2 4xy+= (Shown) 6. Find the exact value of x for which 14(2 ) 33 5(2 )xx− += . [5] 14(2 ) 33 5(2 )xx− += Let 2xu= 2 14 33 5 5 33 14 0 (5 2)( 7) 0 uu uu uu += − − = + − = 2 ( ); 75 ln 727 ln 2 x u NA u x =− = =  =

7. Find the exact value of x such that 2e 6 exx=+ . [4] 2 6xxee =+ Let y = ex, y2 – y – 6 = 0 (y + 2)(y – 3) = 0 y = -2 or y = 3 ex = -2 or ex = 3 (Rejected since ex > 0) x = ln3 8. Use an algebraic method to solve the simultaneous equations 2 23 11 32 ,4 3 1. x y xy − + +  = = [5] ( ) ( ) 23 1 2 2 3 5 1 1 324 22 4 6 5 5 5 4 1 .......(1) x y xy xy yx − + − − +  = = − + = + += 2 2 2 31 0 ......(2) xy xy xy + =  + =  =− Sub (2) into (1): ( )( ) 2 2 1 4 1 16 5 4 1 4 5 1 0 4 1 1 0 1or 1 yy yy yy y y x x −= − + = − − = = =  =− =−

9. Solve the simultaneous equations 2x – 5y = 3 and 2x – 3 = 21 – 5y – 2. [4] Let A = 2x , B = 5y A – B = 3 … (1) 420082525218 =+−= BABA …. (2) Solving (1) & (2), A = 128 , B = 125 Hence 2x = 128 = 27  x = 7 and 5y = 125 = 53  y = 3 Section 2: Logarithmic Functions 1. (a) It is given that 9logwx= . Find 1 x , in terms of w, [2] (b) Given that 4ln 3ln ln 2 ln ln54x y x+ − = − , find the value of x y . [4] (a) 9 1log 9 9 www x x x −=  =  = (b) 4ln 3ln ln 2 ln ln54x y x+ − = − 34 ln ln2 54 xy x   =        34 2 54 xy x= 3 33x y = 3x y=

2. Solve the equation: 2 2 23 log log ( 3) log ( 3)x x x+ = + + − [3] ( ) ( ) 2 2 2 22 2 3 log log ( 3) log ( 3) log 8 log ( 3)( 3) 8 ( 3)( 3) 8 9 0 x x x x x x x x x xx + = + + − = + − = + − − − = 2 ( 9)( 1) 0 91 log , 0 so x= 1 is not valid2 9 xx x or x x x x − + = = =−  − = 3. (i) Show that 77log (7 ) 1 logxx−= . [2] (ii) Hence solve the simultaneous equations 31 13 27 9 y x− =  and 77log (7 ) 1 log (3 5)xy− = + . [2] (i) 7 77 7 7 log (7 ) 1 log 7 log 7 7log 7 log (Shown) x x x x − =− = = (ii) ( ) 31 3 1 3 2 3 1 3 2 13 27 9 3 3 3 33 y x yx xy − −− −− =  = = 3 2 4xy+= --------- (1)

77 77 log (7 ) 1 log (3 5) log log (3 5) 3 5 ---------- (2) xy xy xy − = + =+ =+ Solving (1) and (2) 2 and 1xy= =− 4. Solve the equations, leaving your answers in 3 significant figures where necessary. (a) 5lg)1lg(2 =−x , [3] (b) yy 75 1 =+ . [2] (a) 5lg)1lg(2 =−x 5lg)1lg( 2 =−x 5)1( 2 =−x 5122 =+− xx 0422 =−− xx 2 )4)(1(4)2(2 2 −−−=x 2 542 =x 24.3=x or 24.1−=x (reject) because lg (-ve) is undefined (b) yy 75 1 =+ 7lg5lg)1( yy =+ 5lg 7lg1=+ y y yy 209.11=+ 78.4=y

5. Given that px=2log and qy =2log , express xy2log in terms of p and q. [1] q yxy p x )2( loglog 22 = = Section 3: Word Problems 1. Damian decides to deposit a fixed sum of money into his new bank account at the start of each year. He also decides that he will not draw any money out of the account, but just leave it, and any interest, to build up. At the end of n years, when the interest for that year has been added, he will have a total of $ ( )5200 1.04 1−n . (i) How much will there be in Damian’s account at the end of 9 years? Give your answer to the nearest dollar. [1] (ii) After how many complete years will Damian have, for the first time, at least $3500 in his account? [3] (i) ( ) 95200 1.04 1 $2201.22 $2201− =  (ii) ( )5200 1.04 1 3500−n 35001.04 1 5200 3500lg1.04 lg 1 5200 3500lg 15200 lg1.04 13.1 + +   +  n n n n Ans: 14 years

2. A biologist introduced a batch of 25 butterflies into a garden for breeding. The butterfly population, P, after t weeks, is given by 5e ktPA=+ , where A and k are positive constants. After 12 weeks, there are 31 butterflies. (i) Show that 20A= . [2] (ii) Determine the value of k. [2] (iii) Find the number of weeks it takes for the population to increase by 60% from the beginning. Leave your answer to the nearest integer. [3] (iv) Sketch the graph of P against t. [2] 2i (0) 5e 25 5 e 25 5 20 kt k PA A A A =+ =+ =+ = ii (12) (12) (12) 5 20e 31 5 20e 26 20e e 1.3 12 ln(1.3) 0.021863 0.0219 (3sf) kt k k k P k k =+ =+ = = = == iii 0.021863 0.021863 0.021863 1.6 25 40 40 5 20e e 1.75 ln e ln1.75 0.021863 0.55961 25.5964 26 weeks t t t t t = =+ = = = == iv t P O 25

3. Desert locusts are a certain species of short -horned grasshoppers that are usually found in the deserts of Eutoria. When a drought begins, they start to breed quickly and destroy crops. The population P of the locusts (in thousands) after t weeks of drought, is modelled by the equation 0.510e tPk=+ , where k is a positive real number. (i) Given that the population of the desert locusts is 77 880 after 2 weeks, show that 50.7k = , correct to 3 significant figures. [2] (ii) Using a non-calculator method, (a) find the initial population of the desert locusts, [2] (b) determine the number of complete weeks for the population of desert locusts to first exceed 5 million. [3] (iii) Sketch, in the context of this question, the graph of P against t, stating the coordinates of any points of intersection with the axes. [1] A pest control company has been employed by the government of Eutoria to tackle the problem caused by the desert locusts. The company claims that the population Q of the locusts (in thousands) after t days of administering a biological pesticide is modelled by the equation 2 0.3100e 44tQ −=− . (iv) Determine the number of complete days for the population of the desert locusts to first fall below 1000. [2] (v) Comment on the suitability of the company’s model, justifying your answer. [1] (i) At 2t= ( )0.5 2 10e 77.88k+= 50.697 50.7k = (shown) (ii) (a) At 0t= 010e 50.7 60.7P= + = Initial population is 60,700 (ii) (b) 0.510e 50.7 5000tP= +  0.5e 494.93t  0.5 ln 494.93t 12.409t Number of weeks is 13