Logs
1. Write down the value of each of these
(a) log2 4
(b) log3 27
(c) log3 81
(d) log1 1000
(e) log2 32
2. Find the value of each of the following
(a) log8 16
(b) log9 27
(c) log16 32
(d) log 1 8
2
(e) log 1 81
3
3. Change each of the folowing to index form and solve the equation
(a) log 1 27 = x
3
(b)
log√
2
4=x
(c) log8 x = 2
(d) log64 x =
1
2
4. Solve each of the following equations
(a) log2 x = −1
√
(b) log3 27 = x
(c) logx 2 = 2
(d) log2 (0.5) = x
5. Simplify each of the following expressing your answer without logarithms
(a) log4 2 + log4 32
(b) log6 9 + log6 8 − log6 2
(c) log6 4 + 2 log6 3
6. Write each of the following in the form logy x and then simplify
1
(a) log3 2 + 2 log3 3 − log3 18
(b) log8 72 − log8
9
8
7. If log10 3 = x and log10 5 = y, express in terms of x and y
(a) log10 15
(b) log10 75
9
(c) log10 ( 25
)
(d) log10 1 23
√
(e) log10 45
8. If log3 5 = a, find in terms of a
(a) log3 15
(b) log3 ( 53 )
(c) log3 8 13
(d) log3 ( 25
)
27
(e) log3 75
9. Solve for x
(a) log2 (3x + 1) = 2
(b) log2 (x − 1) = 3
(c) log3 (5x + 2) = 3
(d) log5 (8x + 1) = 2
10. Solve for x
(a) log3 (2x + 5) − log3 (x + 8) = 1
(b) log3 (10x + 7) − log3 (x + 1) = 2
(c) log2 3 + log2 (x + 1) = log2 (x + 11)
(d) log5 (x + 1) = log5 (7x + 1) − 1
11. Solve for x
(a) log4 (3x + 2) =
log2 (x−1)
,x
2
∈R
(b) log5 (2x + 1) + log125 (2x + 1) = 16, x ∈ R
(c) log16 x − log4 x + 3 = 0, x ∈ R
(d) log27 (x + 2) + log3 (x + 2) = 4, x ∈ R
(e) log27 (2x + 4) + log81 (6x + 12) = 56 , x ∈ R
12. Solve for x
(a) log4 (3x + 1) = log2 (x − 1)
(b) log5 (x + 1) + log4 (x + 1) = 3
(c) log8 x − log9 x = 1
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(d) log2 x + log3 x = 1
13. Solve each of the following equations. Give your answers correct to two decimal
places.
(a) 22x − 8(2x ) + 15 = 0
(b) 32x = 24 − 5(3x )
(c) 102x − 5(10x ) = 0
(d) (7x+1 )2 = 100
(e) 32x+1 + 5(3x ) − 2 = 0
14. 9x − 2(3x+1 ) + 8 = 0
15. Solve each of the following equations. Give your answer coorect to two significant
figures.
(a) 32x+1 − 13(3x ) + 14 = 0
(b) 9(52x ) = 4(3(5x ) − 1)
(c) 22x+1 = 12 − 5(2x )
(d) 62x − 7(6x ) + 10 = 0
(e) 2x+1 5x+1 − 9(2x 5x ) − 2 = 0
In the following six questions solve for x and y ∈ R
16. log (5x − y) = log 9 and log(3x + 2y) = log 8
17. log2 (2x + y) = 3 and log2 (3x − 4y) = 0
18. log3 (3x − y) = log3 (y + 1) and log3 2 + log3 (x + y) = 2
19. log x2 = log y and log (2x + y) − log 3 = 0
20. log2 2 + log2 (x + 1) = log2 y and log2 x + log2 y = 2, x > 0, y > 0
21. log2 x − log2 2 = log2 (1 − y) and log2 x + log2 (x + 2y) = 3
The following six questions require changing the base. Solve each for x ∈ R
22. log2 x = log4 (x + 6)
23. log2 (x − 1) = log4 (4x − 7)
24. log3 x + 3 logx 3 = 4
25. log4 x + 2 logx 4 = 3
26. log5 x − 1 = 6 logx 5
27. 4 logx 2 = log2 x + 3
28. If log4 ( xy ) = 5, x, y > 0, find the value of log 2x − log 2x + 3
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29. The point a(p, k) lies on the curve with equation y = log2 x
The point b(q, k) lies on the curve with equation y = log 4x
Find a relationship between p and q and hence evaluate p when q =
9
16
30. log a2 + 2 loga x = loga (5x − 2a) + 1
Write a quadratic equation in terms of x and find. in terms of a, the values of x
31. If log4 xy = 2, prove that log2 x + log2 y = 4
Solve the simultaneous equations Let log4 xy = 2 and (log2 x)(log2 y) = 3
(Hint: Let log2 x = p and log2 y = q
32. If log4 a = k, express the following in terms of k.
(a) log4 a2
(b) log4 4a2
(c) log16 a
(d) loga 4
(e) logk
1
4
The following 8 questions require a calculator. Solve each for n, correct to four
significant figures.
33. 3n = 2500
34. 5n = 680
35. 4n = 20
36. 2n = 31
37. 4n+2 = 3460
38. 32n−1 = 4800
39. 5n−1 = 2n
40. 52n−1 = 4n+1
41. Solve for x
(a) x = loge e
(b) loge x = 2
(c) loge
1
e
=x
√
(d) loge 3 e = x
42. Solve for x
(a) eloge x = 5
(b) loge e4 = x
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(c) eloge 2 = x
(d) 2 loge x = 1 − loge 7
43. Solve the following equations
(a) 3 = e2x
(b) 2 =ln 3x
1
(c) ln x− 5 = 1
(d) 7e7x = 1
44. Solve the following equations
(a) eln5x = 10
(b) ln e3x + 4ln e2x = 7
(c) ex + e−x = 2
(d) 52x =
1
3
45. Evaluate each of the following
(a) ln e
(b) ln e3
(c) ln ( e12 )
√
(d) ln e
46. By taking the log of both sides, verify that if:
(a) e[ x] = a, then x = ln a
(b) elnx = y, then x = y
47. Solve each of the following
(a) ex = 2
(b) ex = 5
(c) ex = −4
(d) ex =
1
3
(e) ex = −1
(f) e2x = 3
(g) ln x = 1
(h) ln x = 2
(i) ln x =
1
2
(j) ln x = −1
(k) ln x = −3
(l) ln x =
1
2
48. By writing ex = y or otherwise, solve each of the following equations
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(a) ex − 5 + 6e−x = 0
(b) e2x − 8ex + 15 = 0
(c) e2x − 3ex + 4 = 0
(d) 3ex − 7 + 2ex = 0
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