The logarithmic function — Paper 4 · IGCSE Mathematics - International 0607

10 y 3 x 0 90 180 270 360 –3 f x = 2 sin x + cos x for 0° G x G 360 ° ^ h g x = 2 - log x for 0° G x G 360° ^ h (a) On the diagram, sketch the graph of y = f x [3] ^ h. (b) On the same diagram, sketch the graph of y = g x [2] ^ h. (c) Solve the equation. 2 sin x + cos x = 2 - log x … [3]

8 marks

Mark scheme: 10(a) Correct Graph 3 M1 for sine graph with one max and one min A1 for x-intercepts at 150 and 330 (approx.) y f(x)=2sin(x)+cos(x) 3 f(x)=2-log(x) A1 for positive y-intercept x 90 180 270 360 3 10(b) Correct Graph with second 2 M1 for correct shape intersection with other graph (if correct) below x-axis 10(c) 6.18 or 6.175... 3 B1 for each 159 or 158.5 to 158.6 320 or 320.3 to 320.4

This question in 0607/41 May/June 2017

6 y 6 x 0 10 f(x) = x - 5 log x (a) On the diagram, sketch the graph of y = f(x) for 0 1 x G 10 . [2] (b) Find the co-ordinates of the local minimum point. ( … , … ) [2] (c) Find the range of f(x) for the domain 1 G x G 5 . … [2] (d) Solve the equation f(x) = 2. x = … or x = … [2] (e) Solve the inequality f(x) 1 2. … [1] (f) (i) Find f(0.001), f(0.000 01) and f(0.000 000 1). f(0.001) = … , f(0.000 01) = … , f(0.000 000 1) = … [1] (ii) Complete the statement. The y-axis is … to the graph of y = f(x). [1]

11 marks

Mark scheme: 6(a) Correct sketch 2 B1 for correct shape 6666 5555 4444 3333 2222 1111 0000 0000 2222 4444 6666 8888 10101010 6(b) (2.17, 0.488) or (2.171…, 0.4877…) 2 B1 for each 6(c) 0.488 - f ( x ) - 1.51 2 FT their 0.488 or 0.4877... - f ( x ) - 1.505... B1 for 0.488 - f ( x ) oe or f ( x ) - 1.51 oe 6(d) 0.502 or 0.5015… 2 B1 for each 5.83 or 5.827… 6(e) 0.502 < x < 5.83 1 FT their (d) or 0.5015... < x < 5.827... 6(f)(i) 15.[0] or 15.00… 1 25.[0] or 25.00… 35. [0] or 35.00… 6(f)(ii) [an] asymptote oe 1

This question in 0607/43 May/June 2017

5 y 2 1 x 0 180 360 540 −1 −2 J N 1 f (x) = sin x° g (x ) = log KK OO sin x° L P (a) (i) On the diagram, sketch the graph of y = f (x) for 0 G x G 540 . [2] (ii) Write down the range of f ()x for 0 G x G 540 . … [1] (b) (i) On the same diagram, sketch the graph of y = g (x) for values of x between 0 and 540. [2] (ii) Give a reason why there are no values of g ()x for 180 G x G 360 . … [1] (iii) Write down the co-ordinates of the minimum points on the graph of y = g (x) . ( … , … ) and ( … , … ) [2] (iv) Write down the equations of the four asymptotes to the graph of y = g (x) . … , … , … , … [2] (c) (i) f (k) = g (k) and 0 G k G 90 . Find the value of k. k = … [1] (ii) Solve the inequality f (x) 2 g (x) for values of x between 0 and 540. … [2] (iii) j is an integer. The equation f ()x = j has no solutions. The equation g ()x = j has no solutions. Write down a possible value of j. j = … [1]

14 marks

Mark scheme: 5(a)(i) Correct2222 sketch 2 B1 for sine graph with different amplitude and/or period but must go through (0, 0) 1111 or for correct sine graph but only one cycle 0000 0000 100100100100 200200200200 300300300300 400400400400 500500500500 -1-1-1-1 -2-2-2-2 5(a)(ii) −-1 f ( x ) - 1 1 5(b)(i) Correct2222 sketch 2 i.e. Correct shape with 2 branches above x- axis and gap of at least 120 between the 1111 branches and only slightly crossing either 0000 0000 100100100100 200200200200 300300300300 400400400400 500500500500 axis. -1-1-1-1 B1 for 2 branches above x-axis but gap less -2-2-2-2 than 120 between the branches and only slightly crossing either axis or one branch correct 5(b)(ii) logarithms of negative numbers do 1 not exist oe 5(b)(iii) (90, 0), (450, 0) 2 B1 for each 5(b)(iv) x = 0, x = 180, x = 360, x = 540 2 B1 for 2 or 3 correct 5(c)(i) 23.5 or 23.51 to 23.52 1 5(c)(ii) 23.5 < x < 156.5 2 B1 for each 383.5< x < 516.5 Allow 23.51 to 23.52, 156.48 to 156.49 Allow 383.51 to 383.52, 516.48 to 516.49 5(c)(iii) Any integer less than – 1 1

This question in 0607/41 May/June 2018

2 y 3 0 x 5 x + 1 (a) On the diagram, sketch the graph of y = log for 0 1 x G 5 . [2] b x l x + 1 (b) Write down the equations of the asymptotes to the graph of y = log b x l. … … [2] x + 1 (c) Solve the equation log = 0. 5 . b x l x = … [1] x (d) On the same diagram, sketch the graph of y = for 0 1 x G 5 . [1] 2 x + 1 x (e) Solve the equation log = . b x l 2 x = … [1] x x + 1 (f) On your diagram, shade the region where y G 0.5 , y H and y H log [1] 2 b x l.

8 marks

Mark scheme: 2(a) Correct sketch 2 Must not cross axes 1111 0.80.80.80.8 0.60.60.60.6 0.40.40.40.4 B1 for correct shape 0.20.20.20.2 1111 0000 0000 1111 2222 3333 4444 5555 2(b) y = 0, x = 0 2 B1 for each If 0 scored, SC1 for answers x-axis and y-axis 2(c) 0.462 or 0.4624 to 0.4625 1 2(d) Correct sketch 1 3333 2.52.52.52.5 2222 1.51.51.51.5 1111 0.50.50.50.5 0000 0000 1111 2222 3333 4444 5555 2(e) 0.742 or 0.7415 to 0.7416 1 2(f) Region that is below y = 0.5 and 1 above other two graphs.

This question in 0607/42 May/June 2019

12 f x = 10 - x g x = x 2 + 1 h x = j x = log 3 x x (a) Find g(3). … [1] (b) Find f(h(2)). … [2] (c) Find g(f(x)) in the form ax 2 + bx + c . … [3] (d) For some functions, p-1(x) = p(x). Write down which two functions, f(x), g(x), h(x) or j(x), have this property. … and … [2] 1 (e) Write h x - as a single fraction in its simplest form. ` j f x ` j … [3] (f) (i) Find j(243). … [1] (ii) Find x when j(x) = 1.5 . x = … [1] (iii) Find j-1(x). j – 1(x) = … [2]

15 marks

Mark scheme: 12(a) 10 1 12(b) 9.5 oe 2 1 1 M1 for 10 − soi e.g. 10 – x 2 12(c) x 2 − 20 x + 101 3 M1 for (10 −x ) 2 + 1 B1 for 100 – 10x – 10x + x2 oe 12(d) f(x) and h(x) 2 B1 for each 12(e) 10 − 2 x 3 M1 for common denominator x(10 – x) oe oe B1 for (10 – x) – x oe seen x (10 − x ) 12(f)(i) 5 1 12(f)(ii) 3 1 3 3 oe or 3 2 or 5.2[0] or 5.196... 12(f)(iii) 3x 2 M1 for x = log 3 y or x = 3 y

This question in 0607/42 May/June 2019

9 (a) Solve the following equations. 135 (i) = 5 x x = … [1] (ii) 3x + 5 = 7x + 25 x = … [2] (iii) 8x 2 = 11 - 2x x = … or x = … [4] (b) Solve the following inequalities. (i) 6 - 2x H 10 … [2] 1 (ii) 2 3 x - 2 … [3] (c) Solve the simultaneous equations. You must show all your working. 3x + 5y =-3 5x - 2y = 26 x = … y = … [4] (d) Solve the equation. log x + 4 log 2 = log 13 x = … [3]

19 marks

Mark scheme: 9(a)(i) 27 1 9(a)(ii) –5 2 M1 for 5 − 25 = 7 x − 3x or better 9(a)(iii) 1.05 or 1.054… 4 2 −±2 2 −×4 8 ×−11 –1.3[0] or –1.304… M3 for 2 × 8 or correct sketch which would lead to solution. b 2 or M2 for correct or b − 4 ac correct 2a or M1 for 8 x 2 + 2 x − 11 or − 8 x 2 − 2 x + 11 or sketch of 8 x 2 or 11 − 2x 9(b)(i) x - − 2 oe 2 M1 for 6 − 10 . 2x or − 2 x . 10 − 6 or 3 − x . 5 or better If 0 scored SC1 for x . − 2 or x = –2 9(b)(ii) 1 3 7 2 < x < 2 oe M2 for x = 2 and x = 3 3 or correct sketch which would lead to solution. or M1 for 1 > 3( x − 2) or better or sketch of 1 y = x − 2 1 or B1 for x < 23 or for x > 2 9(c) Correctly equating one set of M1 coefficients oe Correct method to eliminate one M1 variable [x=] 4 B1 [y=] –3 B1 If 0 scored SC1 for correct substitution into one of original equations and evaluation to find other variable. 9(d) 13 3 4 or 0.8125 M1 for log2 or better 16 p M1 for correct use of log p − log q = log q or use of log p + log q = log pq

This question in 0607/43 May/June 2019

11 (a) Simplify. a 5 # a 4 (i) 3 a … [2] (ii) log 5 (5x ) … [1] (iii) log 9 (3x ) … [1] (b) Solve. 3 log 10 - 2 log 5 = logx x = … [2]

6 marks

Mark scheme: 11(a)(i) a6 final answer 2 B1 for a9 or a2 × a4 or a5 ×a[1] 11(a)(ii) x 1 11(a)(iii) 1 1 x oe 2 11(b) 40 2 M1 for one correct use of alog b = log ba or for correct use of loga – logb = log(a ÷ b)

This question in 0607/43 Oct/Nov 2019

6 (a) y 1.5 0 x 5 (i) On the diagram, sketch the graph of y = log x for 0 1 x G 5 . [2] (ii) Solve the equations. (a) logx = 0.2 x = … or x = … [2] x (b) logx = 1 - 4 x = … or x = … [4] (b) y 1.5 x – 5 0 5 – 5 (i) On the diagram, sketch the graph of y = log x for values of x between - 5 and 5. [2] (ii) Solve the equation log x = 0. 2 . x = … or x = … [2] (c) Write down the range of values of x for which the graph of y = log x is the same as the graph of y = log x . … [1]

13 marks

This question in 0607/41 May/June 2020

11 (a) Solve the equations. (i) 5 + 2x = 1 x = … [2] 10 (ii) 6 - = 1 x x = … [2] (iii) 3 ( 1 - 2x) = 2 - 4 ( x - 7) x = … [3] (b) (i) Solve 6x 2 = 7 - 3 x . Give your answers correct to 3 decimal places. You must show all your working. x = … or x = … [4] (ii) Solve 6y 4 = 7 - 3 y 2 . Give your answers correct to 3 decimal places. y = … or y = … [2] (c) Solve 2 log x + log 5 = 1. x = … [4]

17 marks

Mark scheme: 11(a)(i) –2 2 5 1 M1 for 2 x = 1 − 5 or + x = 2 2 11(a)(ii) 2 2 10 M1 for − = 1 − 6 oe or 6 x − 10 = x x 11(a)(iii) –13.5 3 M1 for correct expansion 3 − 6 x = 2 − 4 x + 28 M1 for correct collection of their terms 3 − 30 = 6 x − 4 x their (3 − 30) M1 for their (6 − 4) 11(b)(i) 6 x 2 + 3 x − 7 = 0 B1 Correct sketch M2 M1 for any U-shaped parabola OR OR b −±3 32 − 4 × 6 × −7 for or b 2 − 4 ac correct 2 a 2 × 6 0.859, –1.359 B1 11(b)(ii) 0.927, 2 FT their (b)(i) –0.927 B1 for each 11(c) 1.41 or 1.414… cao 4 M3 for 5 x 2 = 10 or M2 for log5x 2 [=1] or M1 for logx2 + log5 [=1]

This question in 0607/42 May/June 2020

12 Solve the equations. 2 (a) 6 - =- 2 x x = … [3] (b) 3 + 2 ( 4x + 5) = 1 - 2 ( x + 8) x = … [3] (c) 3 log x + 2 log 3 = 2 log 6 + log 2 x = … [3] (d) 2 x = 10 x = … [3]

12 marks

Mark scheme: 12(a) 0.25 oe 3 M2 for 8 x = 2 or − 2 = −8 x or better −2 or M1 for 6 x − 2 = −2 x or = −8 oe x OR M2 for correct sketch that could lead to correct answer or M1 for appropriate but incomplete 2 sketch e.g. 6 −x 12(b) –2.8 oe 3 M2 for 8 x + 2 x = 1 − 16 − 3 − 10 oe or M1 for 3 + 8 x + 10 or 1 − 2 x − 16 12(c) 2 3 M1 for log x 3 or log32 or log6 2 or better M1 for correct use of p log p − log q = log q or log p + log q = log pq 12(d) 3.32 or 3.321 to 3.322 3 log10 1 B2 for or log 2 10 or log2 log2 or M1 for x log2 = log10 OR M2 for correct sketch that could lead to correct answer or M1 for appropriate but incomplete sketch e.g. y = 2 x

This question in 0607/41 Oct/Nov 2020

7 (a) Solve the simultaneous equations. You must show all your working. 7x + 2y = 8 2x - 3y = 13 x = … y = … [4] (b) Solve. (i) 3x - 4 =- 19 x = … [2] (ii) 15 - 5x = 7 - 3x x = … [2] 28 (iii) =- 4 ( x + 1) x = … [2] (c) 3 log p - log q - log 8 = 2 log x Find x in terms of p and q. x = … [3]

13 marks

Mark scheme: 7(a) Correctly equating one set of coefficients M1 or making x or y the subject of one equation Correct method to eliminate one variable M1 May be intersection of two straight line graphs [x =] 2 A2 A1 for each [y =]–3 If 0 scored for whole question, SC1 for answers that satisfy one equation 7(b)(i) –5 2 M1 for 3x = 4 – 19 oe 7(b)(ii) 4 2 M1 for 5x – 3x = 15 – 7 oe 7(b)(iii) –8 2 M1 for 28 = –4(x + 1) oe 28 or = − ( x + 1) oe 4 7(c) 3 3 M1 for log p3 or log x2 or better p oe final answer M1 for correct use of 8q log a + log b = log ab a or log a − log b = log b

This question in 0607/41 May/June 2021

11 (a) Using a suitable sketch, solve 5 x = 10 . y 12 0 x – 1 3 – 1 (b) Solve. 5 + x 6x - 1 = 2x + 3 You must show all your working. x = … or x = … [5]

8 marks

Mark scheme: 11(a) Correct sketch M2 or M1 for exponential graph 1.43 or 1.430 to 1.431 B1 11(b) Algebraic method (6 x − 1)(2 x + 3) = [5 + x ] or better M1 12 x 2 + 15 x −=8 0 A1 2 M1 Correct use of formula −15 ± 15 − 4 × 12 × −8 x = or correct sketch of parabola 2 × 12 0.403 or 0.4032… B1 –1.65 or –1.653… B1 11(b) Graphical method (1) Correct sketch of y = 6x – 1 M1 5 + x M2 or M1 for hyperbolic graph Correct sketch of y = 2 x + 3 0.403 or 0.4032… B1 –1.65 or –1.653… B1 11(b) Graphical method (2) Correct sketch M3 M2 for parabola y = (6x –1)(2x + 3) oe or M1 for parabola and M1 for y = 5 + x 0.403 or 0.4032... B1 –1.65 or –1.653... B1

This question in 0607/41 May/June 2021

11 (a) f(x) = 3x + 2 g(x) = x2 h(x) = 2x (i) Find f(2). … [1] (ii) Find f(g(3)). … [2] h ( g ( 3)) (iii) Find the value of . g ( h ( 3)) … [3] (iv) Find f -1 ( )x . f -1 ( )x = … [2] (v) Find h -1 ( )x . h -1 ( )x = … [2] 1(b) (i) Find the value of log 3 81 - log 9 3 b l. … [2] 2 (ii) log b 25 = 3 Find the value of b. b = … [2]

14 marks

Mark scheme: 11(a)(i) 8 1 11(a)(ii) 29 2 M1 for g(3) = 9 or 3( x 2 ) + 2 11(a)(iii) 8 3 B1 for h(g(3)) = 29 or 512 B1 for g(h(3)) = 26 or 64 11(a)(iv) x − 2 2 y 2 oe final answer M1 for x = 3y – 2 or y − 2 = 3 x or = x + 3 3 3 11(a)(v) log x 2 M1 for x = 2y or x log2 = log y oe log 2 x or log 2 11(b)(i) 1 2   1  1 = 4 oe B1 for [log 3 81 = ]4 or  log 9    − 2   3   2 11(b)(ii) 125 2 2 B1 for 25 = b 3 oe

This question in 0607/43 May/June 2021

5 (a) X = 3A + 5B Work out the value of B when X = 48 and A = 4. B = … [2] (b) Solve 6 ( 1 - 2x) = 2 + 4 ( x - 1) . x = … [3] 3x - 2 3 + 2x (c) Solve = - 2 . 5 4 x = … [3] (d) Solve 4 log 2 - 2 log x + log 4 = 2 . You must show your working. x = … [4] (e) Solve x = 16 - 6x 2 . Give your answers correct to 2 decimal places. … [3]

15 marks

Mark scheme: 5(a) 7.2 oe 2 M1 for 48 = 3 +4 5B 5(b) 0.5 oe 3 M1 for 6 − 12 x or 2 + 4 x − 4 M1 for correctly collecting their terms e.g. −12 x − 4 x = 2 − 4 − 6 oe 5(c) –8.5 oe 3 M1 for eliminating fractions M1 for expanding brackets and collecting their terms M1 for correctly solving their equation of the form ax = b Max 2 marks for incorrect answer 5(d) 0.8 oe 4 B1 for 2 = 2log10 or log100 M1 for a correct use of log a + log b = log ab a or log a − log b = log b M1 for a correct use of log a b = b log a 5(e) x = –1.72 3 M2 for sketch indicating correct roots x = 1.55 2 −1 1 −−4 6 ( 16) or x = 2  6 or M1 for 6 x 2 + x − 16 [ = 0] or reverse signs If 0 scored, SC1 for one correct answer

This question in 0607/42 Feb/March 2023

11 (a) (i) Write 0.000 021 in standard form. … [1] (ii) Calculate 7.3 # 10 -11 # 4.7 # 10 -7 giving your answer in standard form. ` j ` j, … [1] (iii) Calculate .32 # 10 -200 ' 4 # 10 - 100 giving your answer in standard form. ` j ` j, … [2] 2 (iv) Simplify 5 # 10 p , giving your answer in standard form. ` j … [2] (b) y = 10x Write x in terms of y. x = … [1] (c) Solve 7 x = 14 . x = … [1] 1(d) log y = 1 + 3 log x - log w 2 Find y in terms of x and w. y = … [4]

12 marks

Mark scheme: 11(a)(i) 2.1  105 1 11(a)(ii) 3.431  1017 1 11(a)(iii) 8  10101 2 B1 for 0.8  10100 seen 11(a)(iv) 2.5  10 2 p1 2 B1 for 25  10 2 p or 2.5  10  10 2 p seen 11(b) logy or log10 y final answer 1 11(c) log14 1 1.36 or 1.356… or or log7 14 log7 final answer 11(d) 10x 3 10x 3 3  12 10x 3 w 4 M1 log10 soi or or 10 x w or 1 w 1 w 2 2 M1 for log w or log w or log x3 w final answer M1 for correct use of p log p – log q = q or correct use of log p  log q  log pq

This question in 0607/41 May/June 2023

6 (a) Solve. 7x - 5 = 3x + 13 x = … [2] (b) Solve. 4 ( 2x - 3) = 3 ( 1 - 2 x) x = … [3] (c) Solve. 3x + 2 2 = 8 3x + 2 x = … or x = … [3] (d) Solve. 1 - 2 x 2 = 5x - 1 Give your answer correct to two decimal places. x = … or x = … [3] (e) log x = 1 + 4 log y Find x in terms of y. x = … [3] (f) There are 12 balls in a bag, n of them are blue. A ball is taken from the bag at random and replaced. The probability that the ball is blue is p. 6 more blue balls are added to the bag. A ball is taken from the bag at random. The probability that this ball is blue is 2p. Find the value of p. p = … [4]

18 marks

Mark scheme: 6(a) 4.5 oe 2 B1 for 7 x  3x = 13  5 oe 6(b) 15 3 B1 for 8 x  12 3 6 x oe oe 14 M1 for correctly collecting terms in an equation 6(c) 2 3 B2 for 3 x  2 4 oe ,  2 oe 3 or for 3 3 x  2  x  2   0  oe 4 4(3)( 4) or for oe 2(3) or M1 for  3 x  2  2  8  2 oe 6(d) 0.35 –2.85 3 B2 for –2.851 to –2.850 and 0.350 to 0.351 OR M2 for correct sketch indicating both roots 5 5 2  4(2)(  2) or for 2(2) or M1 for 2 x 2  5 x  2   0  or  2 x 2  5 x  2   0  6(e)  x  10 y 4 3 M1 for logy4 B1 for 1 = log10 6(f) 1 4 B3 for n = 3 oe 4 n n  6 or M2 for 212  18 or for 12p = 36p – 6 oe n n  6 or M1 for p  or 2 p  12 18 n n  6 or for and seen 12 18

This question in 0607/43 May/June 2023

11 (a) Solve the equation. 2 log 5 - 5 log 2 = 3 log 4 - 2 logx a Give your answer in the form b, where a, b and c are integers. c x = … [4] (b) Make x the subject of the formula. x y = 2x + 1 x = … [4]

8 marks

Mark scheme: 11(a) 32 2 4 25 64 B3 for log = log oe 5 32 x 2 OR M1 for log p + log q = log pq or p log p − log q = log q M1 for log5 2 or log 2 5 or log 4 3 or log x 2 oe 11(b) y 2 4 M1 for correctly squaring [ x =] oe final answer M1 for correct elimination of fractions 2 1 − 2 y M1 for correct expansion of brackets and collecting terms into form px=q M1 for correct division by a 2-term expression Max 3 marks for incorrect final answer

This question in 0607/41 Oct/Nov 2024