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AS Pure Mathematics · Edexcel 8MA0 · free A-level maths worksheet

Exponentials and logarithms

Graphs of a^x and e^x, laws of logs, solving equations with logs and exponentials, ln x.

12 original questions, 61 marks. Print the worksheet, mark it with the answers sheet, and use the solutions to see every step written out by hand.

Worksheet PDFWorked solutions PDFAnswers PDF

The questions

1

Without using a calculator, find the exact value of

2 log3 6 − log3 4

3 marks
Show answer
2
2

Solve the equation 5x = 18, giving your answer to 3 significant figures.

3 marks
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x = 1.80
3

Find the exact solution of each equation.

  1. e2x − 1 = 7
  2. ln(3x + 2) = 4
4 marks
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a) x = (1 + ln 7)2 b) x = (e4 - 2)3
4

Given that loga 2 = p and loga 3 = q, write each of the following in terms of p and q

  1. loga 18
  2. loga 427
  3. loga (12a)
5 marks
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a) p + 2q b) 2p - 3q c) 2p + q + 1
5

Solve log2(x + 5) − log2(x − 1) = 3,   x > 1

4 marks
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x = 13/7
6
  1. Show that the equation log3 x + log3(x − 6) = 3 can be written as x2 − 6x − 27 = 0
  2. Hence solve log3 x + log3(x − 6) = 3, explaining why one root of the quadratic is not a solution.
5 marks
Show answer
a) log_3 x(x - 6) = 3, x(x - 6) = 27, x2 - 6x - 27 = 0 b) x = 9 (x = -3 is rejected: log_3(-3) is undefined)
7

Sketch the graph of y = ex − 4

On your sketch show the coordinates of any points where the graph meets the coordinate axes, giving exact values, and state the equation of the asymptote.

−4−3−2−1123−5−4−3−2−1123456Oxy
4 marks
Show answer
through (0, -3) and (ln 4, 0); asymptote y = -4
8

Solve the equation 3x + 1 = 52x − 1

giving your answer in the form ln aln b, where a and b are rational numbers, and then to 3 significant figures.

5 marks
Show answer
x = ln 15 / ln (25/3) = 1.28
9
  1. Show that the substitution u = 2x turns the equation 22x − 7(2x) + 12 = 0 into u2 − 7u + 12 = 0
  2. Hence solve 22x − 7(2x) + 12 = 0, giving any non-exact answer to 3 significant figures.
6 marks
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a) 22x = (2x)2 = u2 b) x = 2 or x = 1.58
10

Solve the simultaneous equations

log2 x + log2 y = 6

log2 x − 2 log2 y = 3

6 marks
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x = 32, y = 2
11
  1. On the same axes, sketch the curves with equations y = e2x and y = 6 − ex, showing the coordinates of the points where each curve meets the y-axis and the equations of any asymptotes.
  2. Find the exact coordinates of the point where the curves meet, explaining why there is only one such point.
−4−3−2−112−2−112345678Oxy
7 marks
Show answer
a) y = e2x through (0, 1) with asymptote y = 0; y = 6 - ex through (0, 5) with asymptote y = 6 b) (ln 2, 4); ex = -3 has no solution
12
  1. Prove that log4 x = 12 log2 x for x > 0
  2. Hence solve log2 x + log4 x = 6
  3. Solve log2 x = log4(x + 6), explaining why only one solution is valid.
9 marks
Show answer
a) x = 4y = 22y so log_2 x = 2y = 2 log_4 x b) x = 16 c) x = 3 (x = -2 is rejected)

Every question here is original and written by Robinson Tuition. It follows the Edexcel AS specification and suits AQA and OCR AS maths too; this is not an exam-board paper.

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