Simplify log2(40) - log2(5), giving your answer as an integer.
(Total for Question 3 is 1 mark)
4
Solve 3x = 81.
(Total for Question 4 is 1 mark)
5
Solve 2x+1 = 16.
(Total for Question 5 is 1 mark)
6
Write down the value of ln(e5).
(Total for Question 6 is 1 mark)
7
Simplify log4(8) + log4(2), giving your answer as an integer.
(Total for Question 7 is 1 mark)
8
Solve 52x = 125.
(Total for Question 8 is 2 marks)
9
Solve 23x-1 = 32.
(Total for Question 9 is 2 marks)
10
Find the value of 3log2(4) - log2(8), giving your answer as an integer.
(Total for Question 10 is 2 marks)
11
Solve log3(4x - 1) = 3.
(Total for Question 11 is 2 marks)
12
Solve 4x = 20, giving your answer to 3 significant figures.
(Total for Question 12 is 2 marks)
13
Solve e2x = 15, giving your answer to 3 significant figures.
(Total for Question 13 is 2 marks)
14
A car's value depreciates according to the model V = 16000 * (0.88)t, where V pounds is the value of the car t years after purchase. Find the time taken for the car's value to fall to half of its original value, giving your answer to 3 significant figures.
(Total for Question 14 is 3 marks)
15
The number of bacteria in a culture is modelled by N = 200 * e0.4t, where N is the number of bacteria present t hours after the culture is first observed. Find the time taken for the number of bacteria to first exceed 5000, giving your answer to 3 significant figures.
(Total for Question 15 is 3 marks)
16
Using the substitution y = ex, solve the equation e2x - 4ex - 5 = 0, giving your answer as an exact value of x.
(Total for Question 16 is 3 marks)
17
A curve C has equation y = 2ln(x - 1) + 3, for x > 1. State the equation of the asymptote of C, and find the exact x-coordinate of the point where C crosses the x-axis.
(Total for Question 17 is 4 marks)
18
A scientist believes that two variables x and y are related by the equation y = A*xn, where A and n are constants. Taking logarithms gives log10(y) = n*log10(x) + log10(A). A graph of log10(y) against log10(x) is a straight line passing through the points (0, 1.2) and (2, 3.6). Find the value of n and the value of A, giving A to 3 significant figures.
(Total for Question 18 is 4 marks)
19
An investment grows according to the model V = 4000 * (1.06)t, where V pounds is the value of the investment after t complete years. Find the value of the investment after 10 years, giving your answer to the nearest pound, and find the number of complete years after which the value of the investment first exceeds 9000 pounds.
(Total for Question 19 is 4 marks)
Mark scheme · P6D Pure: Exponentials and Logarithms: Fluency and Exam Drill
Question 1
B1 26 = 64 stated correctly
Answer: 26 = 64
Question 2
B1 3 cao
Answer: 3
Question 3
B1 3 cao
Answer: 3
Question 4
B1 4 cao
Answer: x = 4
Question 5
B1 3 cao
Answer: x = 3
Question 6
B1 5 cao
Answer: 5
Question 7
B1 2 cao
Answer: 2
Question 8
M1 writes 125 as 53 and equates powers, 2x = 3
A1 x = 1.5 cao
Answer: x = 1.5
Question 9
M1 writes 32 as 25 and equates powers, 3x - 1 = 5
A1 x = 2 cao
Answer: x = 2
Question 10
M1 uses the power law to write 3log2(4) = log2(43) = log2(64)
A1 3 cao
Answer: 3
Question 11
M1 rewrites in index form, 4x - 1 = 33
A1 x = 7 cao
Answer: x = 7
Question 12
M1 takes logs of both sides, x ln4 = ln20
A1 awrt 2.16
Answer: x = 2.16 (3 s.f.)
Question 13
M1 takes natural logs of both sides, 2x = ln15
A1 awrt 1.35
Answer: x = 1.35 (3 s.f.)
Question 14
M1 sets up the equation 0.88t = 0.5
M1 takes logs of both sides, t ln(0.88) = ln(0.5)
A1 awrt 5.42
Answer: t = 5.42 years (3 s.f.)
Question 15
M1 sets up the equation 200e0.4t = 5000, leading to e0.4t = 25
M1 takes natural logs of both sides, 0.4t = ln25
A1 awrt 8.05
Answer: t = 8.05 hours (3 s.f.)
Question 16
M1 substitutes to form the quadratic y2 - 4y - 5 = 0
M1 factorises (y - 5)(y + 1) = 0 and rejects y = -1 since ex > 0
A1 x = ln5 cao (exact)
Answer: x = ln5
Question 17
B1 asymptote x = 1
M1 sets y = 0 and rearranges to ln(x - 1) = -1.5
M1 takes exponentials of both sides, x - 1 = e-1.5
A1 x = 1 + e-1.5 cao (exact)
Answer: Asymptote x = 1; curve crosses the x-axis at x = 1 + e-1.5
Question 18
M1 finds the gradient n = (3.6 - 1.2)/(2 - 0)
A1 n = 1.2 cao
M1 identifies the y-intercept log10(A) = 1.2 and forms A = 101.2
A1 awrt 15.8
Answer: n = 1.2, A = 15.8 (3 s.f.)
Question 19
M1 substitutes t = 10 into the model
A1 GBP 7163 cao (nearest pound)
M1 sets up the inequality 1.06t > 2.25 and takes logs to find t > 13.9 (3 s.f.)
A1 t = 14 cao
Answer: Value after 10 years is GBP 7163; the value first exceeds GBP 9000 after 14 complete years