1 Find the exact solutions of the equation . ln ln x x 3 5 2 5 1 0 2 + - = ` j [4]

1 ( )( ) 3ln5 1 ln5 1 0 − + = x x 1 ln5 3 = x , ln5 1 = − x M1 For recognition of a quadratic in ln5x and attempt to solve to obtain ln5 = x k 1 3 1 e , 5 = x 1 3 ln5 3 e , e 5 − oe 1 1 ln5 1 e , , e 5e 5 − −− = x oe 3 Dep M1 for dealing with their ln5 = x k correctly once A1 for 1 3 1 e 5 = x oe isw A1 for 1 5e = x oe isw To earn marks, you must first recognize the equation as a quadratic in ln(5x) and attempt to solve it for ln(5x). Then, correctly convert your ln(5x) solutions into x values using the exponential function. Finally, state both exact solutions for x.

1 Write ( ) p r q pqr r 2 1 3 2 3 - - ` j in the form p q r a b c, where a, b and c are constants. [3]

1 2 0 5 3 − − p q r 3 B1 for 0 = a B1 for 5 = − b B1 for 2 3 = − c

2 (a) Given that xy xz z zy x y z a b c 3 3 = - ` ` j j , find the exact values of the constants a, b and c. [3] (b) Solve the equation ( ) ( ) 5 2 17 2 3 0 p p 2 1 - + = + . [4]

2(a) 10 1 or 3 3 3 = a B1 7 1 or 2 3 3 = b B1 9 1 or 4 or 4.5 2 2 = c B1 2(b) ( ) ( ) 2 10 2 17 2 3 0 − + = p p ( ) ( ) ( ) ( ) 5 2 1 2 2 3 0 − − = p p 1 3 2 , 2 5 2 = = p p M1 For recognition of a quadratic in 2 p , attempt to factorise and solve for 2 p 1 ln ln1.5 5 or ln 2 ln 2 = = p p oe M1 For correct attempt to deal with 2 = p k 2.32 − A1 0.585 A1 (a) To earn these marks, you must correctly identify the values for a, b, and c based on the given matrix equation. (b) You need to recognize the equation as a quadratic in 2^p, factorize and solve for 2^p, then correctly use logarithms to solve for p, providing both answers to 3 significant figures.

3 (a) Write lg lg a b 3 2 4 + - as a single logarithm to base 10. [4] (b) Solve the equation log log a 3 4 2 7 a 4 + = . [5]

3(a) 2 4 1000 lg a b 4 B1 for 3 lg1000 = B1 for use of power rule once B1 for use of addition or subtraction rule once B1 All correct 3(b) Either 4 3 3log 4 log = a a B1 ( ) 2 4 4 2 log 7log 3 0 − + = a a ( )( ) 4 4 2log 1 log 3 0 − − = a a 4 4 1 log or log 3 2 = = a a M1 For obtaining a quadratic equation and solution 1 3 2 4 or 4 = = a a M1 Dep For dealing with the logarithm correctly once, may be implied by a correct solution 64 A1 2 A1 Or 4 2 2log log 4 = a a (B1) ( ) 2 3 log 4 7log 4 2 0 − + = a a ( )( ) 3log 4 1 log 4 2 0 − − = a a 1 log 4 or log 4 2 3 = = a a (M1) For obtaining a quadratic equation and solution 1 2 3 4 or 4 = = a a (M1) Dep For dealing with the logarithm correctly once, may be implied by a correct solution 64 (A1) 2 (A1) (a) To get full marks, you must correctly apply the logarithm power rule, then the addition/subtraction rules, and finally express lg 1000 as 3, combining everything into a single logarithm. (b) You need to correctly apply logarithm rules to form a quadratic equation in log_4 a (or log_a 4), solve this quadratic, and then correctly convert the logarithmic solutions back to values for 'a'.

4 (a) Given that r q pr pq p q r a b c 3 2 3 = - - ` j , find the value of each of the constants a, b and c. [3] (b) Solve the equation x x 3 8 5 0 5 4 5 2 - + = . [4]

4(a) 7 2 = a B1 1 = b B1 1 6 = c B1 4(b) 2 2 5 5 3 5 1 0    − − =          x x 2 M1 for recognition of a quadratic in 2 5 x Dep M1 for solution and a correct attempt to get at least one solution for x 3.59 A1 1 A1 (a) You earn one mark for correctly identifying the value of 'a', one mark for correctly identifying 'b', and one mark for correctly identifying 'c'. (b) The first method mark is for recognizing that the equation is a quadratic in terms of x^(5/2) and attempting to factorize or use the quadratic formula. The second method mark is for correctly solving this quadratic for x^(5/2) and then attempting to find at least one value for x. The final two marks are for providing the two correct numerical solutions for x.

Cambridge IGCSE Additional Mathematics · 0606

IGCSE Additional Mathematics (0606) Logarithmic and exponential functions past paper questions

These questions test the laws of logarithms and exponential models, including solving equations and linearising data to a straight line.

134 free questions from 2016–2021, each with its marking scheme. Open the full topical bank to filter by subtopic, paper, and year.

Logarithmic and exponential functions in IGCSE Additional Mathematics 0606 exams, 2020–2025

Across 2020–2025, Logarithmic and exponential functions accounted for 136 questions worth 1,003 marks in Cambridge IGCSE Additional Mathematics (0606) papers.