A-level · Exponentials & logarithms

The laws of logarithmsExam-style questions

The product, quotient and power laws — turning sums and differences of logs into a single log (and back) — plus log_a a = 1, log_a 1 = 0 and the reciprocal rule.

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12 questions · A-level level · mark scheme and worked solution on each · no sign-up to try

DifficultyAll Level 2 Level 3 Level 4

1
In this question you must show all stages of your working.

Solutions relying on calculator technology are not acceptable.
(a) Using the laws of logarithms, solve

$\log_2(x + 4) + \log_2(x - 7) - \log_2(x - 5) = 2$

(5 marks)
(b) The curve $C_1$ has equation $y = 4^{x}$ and the curve $C_2$ has equation $y = 3 \times 2^{x} + 10$

The two curves intersect at a single point, $P$ .

Find the exact coordinates of $P$ . Give your answer in the form $(\log_2 a, b)$ , where $a$ and $b$ are integers.

(5 marks)
Level 4 · 10 marksTry it
2
In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

Given that $a = \log_3 5$ ,
(a) express each of the following in terms of $a$ , giving each answer in its simplest form.

(i) $\log_3 75$

(ii) $\log_3\left(\dfrac{\sqrt{5}}{27}\right)$

(3 marks)
(b) Find the exact value of $x$ for which

$5^{2x} \times 3^{x + 1} = 45$

Give your answer in its simplest form in terms of $a$ .

(4 marks)
Level 4 · 7 marksTry it
3
(a) Show that the equation

$\log_3(x - 5) + \log_3\left(x^{2} + 3x - 5\right) = 2$

can be written as

$x^{3} - 2x^{2} - 20x + 16 = 0$

(4 marks)
(b) One solution of the equation $x^{3} - 2x^{2} - 20x + 16 = 0$ is $x = -4$

(i) Using algebra, find the other two solutions of $x^{3} - 2x^{2} - 20x + 16 = 0$ , giving each in simplest surd form.

(ii) Hence solve the equation in part (a).

(4 marks)
Level 4 · 8 marksTry it
4
A bicycle factory trains new employees to build wheels.

The mean time, $y$ minutes, that a new employee takes to build one wheel is modelled by

$y = a + b\log_{10} x$

where $x$ is the number of days for which the employee has been building wheels, and $a$ and $b$ are constants.

The factory uses the model for the first $40$ days.

The table shows the mean times for employees on their $4$ th day and on their $36$ th day.
Number of days, $x$ $4$ $36$
Mean time to build one wheel, $y$ (minutes) $40$ $17$
(a) Substituting the values for day $4$ into the model gives

$40 = a + b\log_{10} 4$
(a) (i) Write down the equation given by the values for day $36$

(ii) Show that

$b = -\frac{23}{\log_{10} 9}$

(iii) Find the value of $a$

Give your answer to two decimal places.

(5 marks)
(b) Determine whether the model gives a realistic time for an employee who has been building wheels for one year.

(2 marks)
Level 4 · 7 marksTry it
5
Given that $\log_a 3 = p$ and $\log_a 5 = q$ , express $\log_a 45$ in terms of $p$ and $q$ .
Level 4 · 2 marksTry it