LOGARITHMS EXAM QUESTIONS

LOGARITHMS EXAM QUESTIONS

Question 1 (**) Show clearly that log 36 1 a + log 56 log 48 log 4 a a = a. proof Question (**) Simplify giving the final answer as an integer. log 5 + log1.6, 3 Question 3 (**+) Given that y = x = p and 4 q, show clearly that 3 log ( x y) = 3p + q. proof

Question 4 (**+) Simplify each of the following expressions, giving the final answer as an integer. a) log 3 log 4. b) log a a 1 4loga a, a > 0, a 1. Full workings, justifying every step, must support each answer. 3, 6 Question 5 (**+) Given that y = log x, write each of the following expressions in terms of y. a) log x b) log ( 8x ) y, 3+ y

Question 6 (**+) Given that y = 4 10 x express x in terms of y, giving an exact simplified answer in terms of logarithms base 10. x = 1 log 1 4 10 ( y) Question 7 (**+) An exponential curve has equation x y = ab, x R, where a and b are non zero constants. Make x the subject of the above equation, giving the final answer in terms of logarithms base 10. log y log a x = logb

Question 8 (**+) Solve the following logarithmic equation log10 x + log10 3 = log10 75. x = 5, x 5 Question 9 (**+) Solve the following logarithmic equation log x + log ( x 3) = log 10. a a a CH, x = 5, x

Question 10 (***) An exponential curve C has equation a) Sketch the graph of C. 1 y = 3 x, x R. b) Solve the equation y = 3, giving the answer correct to 3 significant figures. CB, 0.369

Question 11 (***) Given that p = loga 4 and q = loga 5, express each of the following logarithms in terms of p and q. a) log 100 a b) log 0.4 a The final answers may not contain any logarithms. CG, p + q, 1 p q Question 1 (***) Solve the following logarithmic equation log (4t + 7) log t =. 5 5 CL, t = 1 3

Question 13 (***) Given that p = log 3 and q = log 5, express each of the following logarithms in terms of p and q. a) log 45 b) log 0.3 The final answers may not contain any logarithms. MP1-N, p + q, p q 1

Question 14 (***) Solve each of the following equations, giving the final answers correct to three significant figures, where appropriate. a) 7 x = 10. 9 b) log y = log y. CJ, x 1.18, y = 1, 8 8 Question 15 (***) Solve the following logarithmic equation for x. a x a x a log ( 10) log = log 3. x = 10, x 1

Question 16 (***) Solve the following logarithmic equation for x. log x = log 18 + log ( x 4). a a a CC, x = 6, 1 Question 17 (***) Solve the following logarithmic equation log (z + 1) = + log z. CO, z = 1

Question 18 (***) Solve the following logarithmic equation for y. log y log (5y 4) = log 4. a a a x = 8, 1 Question 19 (***) It is given that x satisfies the logarithmic equation where k > 0, a > 0, a 1. log x = (log k log ), a a a a) Find x in terms of k, giving the answer in a form not involving logarithms. Suppose instead that x satisfies ( ) log 5y + 1 = 4 + log 3 x x where x > 0, x 1 and y > 0, y 1. b) Solve the above equation expressing y in terms of x, giving the answer in a form not involving logarithms. k x =, 4 4 3x 1 y = 5

Question 0 (***) Solve the following logarithmic equation log 5(15 x ) = 4. x = 5 Question 1 (***) Solve the following logarithmic equation 1+ log = log 16 3. 5 x 5 ( x ) x = 3, x = 1 5

Question (***) Every 1 invested in a saving scheme gains interest at the rate of 5% per annum so that the total value of this 1 investment after t years is y. This is modelled by the equation y = 1.05 t, t 0. Find after how many years the investment will double. t 14. Question 3 (***) Solve each of the following logarithmic equations. a) log 16 = log 9 +. x b) log 7 = 3 + log 8. y x y x = 4, x 4, y = 3 3 3

Question 4 (***) Solve each of the following equations, giving the final answers correct to three significant figures, where appropriate. x a) 3 = 900. b) log ( 7 y 1) 3 log ( y 1) = +. CP, x 5.56, y = 7 Question 5 (***+) Simplify fully 1+ log 3 + log 4, giving the final answer as a single logarithm. n n log n(36 n )

Question 6 (***+) Solve each of the following exponential equations, giving the final answers correct to 3 significant figures. a) b) 1 300 5 x = 4. y 1 10 + =. y CM, x 130, y 1.16 Question 7 (***+) Solve the following logarithmic equation log ( w + 4w + 3) = 4 + log ( w + w), w 1. w = 1 5

Question 8 (***+) Solve the following exponential equation x 1 1 = 6, giving the answer as single logarithm of base. x = log 6 = 1+ log 3 Question 9 (***+) Solve the following simultaneous logarithmic equations ( ) log xy = 0 ( ) log x y = 3. CU, x = 4, y = 1

Question 30 (***+) Solve the following logarithmic equation log t = 1+ log 7t. 3 3 t = 1, t 0 Question 31 (***+) Solve the following logarithmic equation log 8 3log t = 3. 3 3 CE, t = 3

Question 3 (***+) Solve the following logarithmic equation log (4 w) log w = 1. 5 5 CA, w = 4, w 1 5 Question 33 (***+) Simplify fully the following logarithmic expression, showing clearly all the workings. log ( 10 + 3 10 ) + log ( 10 + 90 + 90 ) + log( 10 90 + 90 ). 1

Question 34 (***+) Solve the following logarithmic equation log y + log (3y + 4) = log (3y 4). y = 4, y 3 Question 35 (***+) Solve the following logarithmic equation log (6 x) = 3 log x. x =, 4

Question 36 (***+) Solve the following logarithmic equation log4 x = log3 9. x = 16 Question 37 (***+) Solve each of the following equations. a) 1 x+ 3 = 3.43. + + = +. b) log ( y ) log ( 4y 3) log ( y 1) 5 5 5 x 0.480, y = 7

Question 38 (***+) The population P of a certain town in time t years is modelled by the equation kt P = A 10, t 0, where A and k are non zero constants. When t = 3, P = 19000 and when t = 6, P = 38000. Find the value of A and the value of k, correct to significant figures. CV, A = 9500, k = 0.10 Question 39 (***+) Solve the following logarithmic equation log x log ( x ) =. 3 3 CF, x = 3, 6

Question 40 (***+) Solve the following logarithmic equation x x 1 3 log 4 = log 7. x = 3 Question 41 (***+) Given that a 0, b 0, y 0 and ( ) + log b + 3log y = log a y, a a a express y in terms of a and b, in a form not involving logarithms. a y = b

Question 4 (***+) x log 1 = log(10 x y) y, x 0, y 0. Find the exact value of y. y = 3 1 100 Question 43 (***+) non calculator The points P and Q lie on the curve with equation y = 6log x log 7, x > 0. The x coordinates of P and Q are 3 and 6, respectively. Find the gradient of the straight line segment PQ.

Question 44 (***+) y = 3 x. a) Describe the geometric transformation which maps the graph of the curve with equation y = x, onto the graph of the curve with equation y = 3 x. b) Sketch the graph of 3 x y =. The curve with equation point P. x y = intersects the curve with equation y = 3 x at the c) Determine, correct to 3 decimal places, the x coordinate of P. CR, vertical stretch by scale factor 3, x 0.79

Question 45 (****) 1 3 x y =, x R. a) Sketch the graph of coordinate axes. 1 3 x y = showing the coordinates of all intercepts with the b) Find to 3 significant figures the x coordinate of the point where the curve 1 3 x y = intersects with the straight line with equation y = 10. c) Determine to 3 significant figures the x coordinate of the point where the 1 curve y 3 x = intersects with the curve y = x. 3.10,.71 Question 46 (****) Solve the following logarithmic equation 16log x + 4log x + log x = 37, x > 0. 4 16 x = 4

Question 47 (****) In 1970 the average weekly pay of footballers in a certain club was 100. The average weekly pay, P, is modelled by the equation P = A b, t where t is the number of years since 1970, and A and b are positive constants. In 1991 the average weekly pay of footballers in the same club had risen to 740. a) Find the value of A and show that b = 1.10, correct to three significant figures. b) Determine the year when the average weekly pay of footballers in this club will first exceed 10000. CQ, A = 100, 019

Question 48 (****) Solve each of the following equations, giving the final answers correct to three significant figures, where appropriate. a) 3 6 x+ = 30. log 1 + 5 log 1 =. b) ( y ) ( y) 4 4 c) t t 8 8 6 = 0. x 0.0339, y = 11 0.393, t 0.58 8

Question 49 (****) Solve the following simultaneous equations, giving your answers as exact fractions log y = log x + 4 y x+ 3 8 = 4. x = 3, y = 4 11 Question 50 (****) Show clearly that log5 6 + log5 log5 9 = 3log5. proof

Question 51 (****) Solve the following logarithmic equation log ( x + 4) + log ( x + 16) = 1+ log x. 10 10 10 x = 4, x 16 9 Question 5 (****) Simplify log4 8 log7 3, giving the final answer as a simplified fraction. 7 6

Question 53 (****) Solve each of the following equations. a) 1 ( ) x 4 3 6 = 1.89. + = +. b) log ( 8y 1) log ( y 1) 3 log ( y 4) CD, x 9.00, y = 4 5 Question 54 (****) Simplify 1 +, log 1 8 log 8 giving the final answer as an integer. 6

Question 55 (****) Given that a = logb16, express log b(8 b ) in terms of a. 3 1 4 a + Question 56 (****) log 1 a y = 3 and log8 a = x + 1. Show clearly that y = 1 x + proof

Question 57 (****) It is given that p = log6 5 and q = log6. Simplify each of the following logarithms, giving the final answers in terms of p, q and positive integers, where appropriate. i. log6 ( 00 ). ii. log6 ( 3. ). iii. log6 ( 75 ). SYN-B, p + 3q, 4q 1 p, p + 1 q

Question 58 (****) Solve the following exponential equation, giving the answer correct to 3 s.f. x x 6 = 0. x 1.58 Question 59 (****) Two curves C 1 and C are defined for all values of x and have respective equations y 1 = 8 x and y = 3 x. Show that the x coordinate of the point of intersection of the two curves is given by 1. 3 log 3 proof

Question 60 (****) Solve the following logarithmic equation log x = log4. x =

Question 61 (****) The functions f and g are defined as f x ( x) ( ) = 3 1, x R, x 0 ( ) log g x = x, x R, x 1. a) Sketch the graph of f. Mark clearly the exact coordinates of any points where the curve meets the coordinate axes. Give the answers, where appropriate, in exact form in terms of logarithms base. Mark and label the equation of the asymptote to the curve. b) State the range of f. c) Find f ( g ( x )) in its simplest form. SYN-B, ( 0, ), ( ) log 3, 0, 1 6 f g x = 1 x y =, 1< f ( x), ( ( ))

Question 6 (****) Solve the following logarithmic equation log3 x = log9 7. x = 3 3 Question 63 (****) The points (,10 ) and ( 6,100 ) lie on the curve with equation y n = ax, where a and n are non zero constants. Find, to three decimal places, the value of a and the value of n. a =.339, n.096

Question 64 (****) Solve the following exponential equation, giving the answer correct to 3 s.f. y y ( ) 4 3 10 = 0. CN, y.3 Question 65 (****) Solve the following logarithmic equation 3 x 9 log log x = 3. x = 9, x 0

Question 66 (****) Show that x = 4 and y = 8 is the only solution pair of the following logarithmic simultaneous equations log (3x + 4) = 1+ log y. log y = 3log x. proof

Question 67 (****) A population P of an endangered species of animals was introduced to a park. The population obeys the equation 15ka P =, t 0, t k + a where k and a are positive constants, and t is the time in years since the species was introduced to the park. Initially 100 individual animals were introduced to the park, and this population doubled after 5 years. a) Show that k = 8. b) Find the value of a, correct to 4 significant figures. c) Determine the value of t when the P = 400. d) Explain why this population cannot exceed 500. t MP1-C, a 1.17, t 14.13

Question 68 (****) Solve the following logarithmic equation log x log x =. 3 9 x = 16, x 0 Question 69 (****) Solve the following logarithmic equation log log 4 = 1. 4 x 16 ( x ) CI, x = 8

Question 70 (****) Solve the following simultaneous equations log x + log y = 4 4 x + y = 10 MP1-G, x =, y = 8 or x = 8, y = Question 71 (****) Two curves C 1 and C are defined for all values of x and have respective equations y 1 = 7 x and y = 5 x. Show that the x coordinate of the point of intersection of the two curves is given by 1. log 7 log 5 SYN-G, proof

Question 7 (****) Solve the following exponential equation, giving the answer correct to 3 s.f. t+ 1 t 1 3 = 6 + 3 SYN-I, t = 1 or t 1.63 Question 73 (****) Solve the following simultaneous logarithmic equations log y x = 5 log x = + log y. Give the answer as exact simplified surds. CK, x = 4, y =

Question 74 (****) Solve the following logarithmic equation log x + log 16 = 3, x > 0, x 1. 4 x x = 4, 16 Question 75 (****) Find, to the nearest integer, the solution of the following exponential equation 1 4 x 500 500 =. x 111

Question 76 (****) Solve the following simultaneous logarithmic equations. 3log ( xy) = 4log 8 x log y = 1+ log x SYN-C, x =, y = 8

Question 77 (****) y y = log x O A B P Q y = log 4 x ( x ) x = k The figure above shows the graphs of the curves with equations y =, and y log ( x 4) log x =. The points A and B are the respective x intercepts of the two graphs. a) Describe the geometric transformation which maps the graph of y = log x onto the graph of y log ( x 4) =. b) State the distance AB. The straight line with equation x = k, where k is a positive constant, meets the graph of y = log x at the point P and the graph of y = log ( x 4) at the point Q. c) Given that the distance PQ is units determine the value of k. CV, translation, 4 units to the "right", AB = 4, k = 16 3

Question 78 (****) The radioactive decay of a phosphorus isotope is modelled by the equation 0.t m = m0, t 0 where m is the mass of phosphorus left, in grams, and t is the time in days since the decay started. The initial mass of phosphorus is m 0. b) Find the mass of the phosphorus left, when an initial mass of 0 grams is left to decay for 10 days, according to this model. m An initial mass, m 0 grams, of this type of phosphorus decays to 0 64 grams in T days. c) Find the value of T. After N days have elapsed, less than 1% of this type of phosphorus remains from its initial mass m 0. d) Find the smallest integer value of N. CX, m = 5, T = 30, N = 34

Question 79 (****) Solve the following logarithmic equation log x log 4 = 1. 4 x CZ, x = 1, 16 4 Question 80 (****) Solve the following simultaneous logarithmic equations. log ( y 1) = 1+ log x log y = + log x. 3 3 MP1-A, x = 1, y = 3 or x = 1, y = 3 4

Question 81 (****) Solve the following logarithmic equation log18 log 8 = log x. log x CV, x = 4, 1 4 Question 8 (****) Two curves C 1 and C are defined for all values of x and have respective equations y 1 = 9 x and y = 6 5 x. Show that the x coordinate of the point of intersection of the two curves is given by : 1+ log3 log 5 3 MP1-K, proof

Question 83 (****+) Solve each of the following equations. a) 1 4 3x+ 1 600 600 =. b) ( y ) log + 5 = 1 log y. 3 3 MP1-I, x = 9.71504..., y = 1

Question 84 (****+) x y = 1 ( ). a) Describe the geometric transformation which maps the graph of the curve with equation x x y = 1. y =, onto the graph of the curve with equation ( ) b) Sketch the graph of 3 x y =. x The curve with equation y = 1 ( ) intersects the curve with equation y = 3 x at the point P. c) Determine, as an exact simplified surd, the y coordinate of P. CZ, reflection in the y axis, y = 3

Question 85 (****+) Solve each of the following logarithmic equations, giving the answers in exact simplified form where appropriate. a) 1 4 log (56 x ) = 1+ log x. y log + log y = 8. b) SYN-E, x = ± 8, y = 16

Question 86 (****+) Solve the following simultaneous logarithmic equations. ( ) log x y = 11+ 1 log y = 3log x. MP1-R, x = 8, y = 1 16

Question 87 (****+) Solve each of the following equations, giving the final answers correct to three significant figures, where appropriate. a) x+ 4 3 = 3 4. x b) log 1 a ( 1 x ) loga ( 9 16x ) + = +. MP1-E, x 8.64, y = 16

Question 88 (****+) Solve the following exponential equation. x+ 1 1 x 4 3 = 4. Give the answer correct to 3 decimal places SYN-K, x 0.855 Question 89 (****+) Solve the following logarithmic equation. ( ) ( ) log x + log x 1 log 5x + 4 = 1. MP1-P, x = 4

Question 90 (****+) The following three numbers log10, log10 ( 1) x, log ( ) 10 3 x +, are consecutive terms in an arithmetic progression. Determine the value of x as an exact logarithm, of base. SP-G, x = log 5

Question 91 (****+) 4 f ( x ) = x Show that the solution of the equation ( 1) 5 f x = is given by 8 1 x = 4log 10 SYN-S, proof

Question 9 (****+) log x log y = 1 log ( 4x y ) = 1. Solve the above simultaneous logarithmic equations, giving the final answers as exact powers of. MP1-X, 1 3 4 x =, y =

Question 93 (****+) Solve the following logarithmic equation log x log x 5 5 + 5 = 30, x > 0. SYN-T, x = 1, x = 10

Question 94 (****+) It is given that 3 3 r r loga x = ( loga x ), r= 1 r= 1 where a and x are positive numbers such that x Show clearly that a, x 1 and a > 1. 1 1 x a ± =. SP-D, proof

Question 95 (****+) ( ) ( ) log x + log x 1 log 5x + 4 = 1. Find the only real root of the above logarithmic equation. SYN-V, x = 4

Question 96 (****+) Solve the following simultaneous logarithmic equations log x y = 100 xy log = 1, 10 given further that x, y R, with x > 0, y > 0. SP-A, x = 10, y = 100, or the other way round

Question 97 (****+) Solve the following logarithmic equation, over the largest real domain x x log 1 =, x R. SP-J, x = 5

Question 98 (****+) Solve the following logarithmic equation log 5 + log 4 x ( 4 x) 4 x + = log 0. MP1-U, x = 1, 1, 1 16 Question 99 (*****) Solve the following logarithmic equation log 7 log 4 4 x x 7 ( x) 4 + log = 0 CS, x =, x = 4, x = 16

Question 100 (*****) Show by valid mathematical arguments that 8 9 8! < 9! SP-T, proof

Question 101 (*****) i. Simplify the following expression. 9log4 + log4 7. Show detailed workings in this simplification. ii. It is given that t 1 t = t ( k ) 5 10 t = k Determine the value of k. SPX-K, 3, k = 1.5

Question 10 (*****) On the 1 st January 000 a rare stamp was purchased at an auction for 16384 and by the 1 st January 010 its value was four times as large as its purchase price. The future value of this stamp, V, t years after the 1 st January 000 is modelled by the equation V t = Ap, t 0, where A and p are positive constants. On the 1 st January 1990 a different stamp was purchased for. The future value of this stamp, U, t years after the 1 st January 1990 is modelled by the equation U t = Bq, t 0, where B and q are positive constants. Given further that q = p, determine the year when the two stamps will achieve the same value according to their modelling equations. SP-V, 01

Question 103 (*****) It is given that a and x are positive real numbers such that x a, x 1 and a > 1. n is a positive integer such that n > 1. Show that the equation can be written as n n r r loga x = ( loga x ), r= 1 r= 1 n ( x) n( n ) x ( n )( n ) log + 1 log + 1 + = 0. a a SP-X, proof

Question 104 (*****) It is given that a log b log a b, a > 0, b > 0. i. Prove the validity of the above result. ii. Hence, or otherwise, solve the following simultaneous equations log y log x + 10 = 7, y 1 x = 10. SYN-X, ( 10000,3 ),( 1000,4 )

Question 105 (*****) Solve the following logarithmic equation. 3 + 8log 1 8 + 4 3 8 4 3 = 0, k > 0, k 1. k SP-F, k = 16

Question 106 (*****) Solve the following logarithmic equation. ( ) ( ) log 1 + log 8 + log 1 = 1, x > 0, x 1. 4 4x x 1 x Give the answers in exact simplified form where appropriate. SPX-F, x = 16, x = ( ± ) 1 7 73

Question 107 (*****) Solve the following simultaneous equations. 1 1 r k k [ a] = ( + b ) and ( 1 b) [ log a] log 1 r= 0 k= 1 r 7 + =. 5 k= 1 r= 0 You may leave the answers as indices in their simplest form, where appropriate. SP-W, [ a b] 13, = 3, 1 = 4, 5 4 5

Question 108 (*****) It is given that a log b log a b, a > 0, b > 0. a) Prove the validity of the above result. b) Hence, or otherwise, solve the equation log x log 3 3 + 3 x = 36. SP-Y, x = 100

Question 109 (*****) It is given that 4 p 1 log 3.6 q = 6 ( ) and q p 1 log ( 75) Solve these simultaneous equations, to show that p = log 6 k, + =. 6 where k is a positive integer to be found. SPX-J, k =

Question 110 (*****) Determine the value of ( ) f 1 1 ( ) 4 x + x 3, x R. log f a, where 10 a =. log 4 log 9 x 10 10 SP-I, f ( a ) = 4

Question 111 (*****) Show that the following logarithmic equation has no real solutions. 4 3 x x x x x + log + 7 + 5 =, x R. SP-L, proof

Question 11 (*****) It is given that for x > 0, x 1 and y > 0, y 1 log x y = log x and log ( x y) = log ( x + y). y x y Show that x 4 x 1 = 0. SP-H, proof

Question 113 (*****) 1 logsin xcos x ( sin x) logsin xcos x ( cos x) =. 4 Show that the solution of the above equation is given by x = 1 π 4 ( 8 n 7 ), n N. CT, proof