Sources: Questions - old Kuen Cheng's Textbook suitable for SPM too

1. 3. 𝑐. log2 3 = 1.585
log2 5 = 2.322
log2 60 = ?
log2 60 = log2 3 × 5 × 22
= log2 3 + log2 5 + 2 log2 2
= 1.585 + 2.322 + 2
= 5.907

2. 3. 𝑔. log2 3 = 1.585
log2 5 = 2.322
log2 0.24 = ?
log2 0.24 = log2
24
100
= log2 3 + log2 2 − 2 log2 5
= 1.585 + 1 − 2 2.322
= −2.059
= log2
6
25
= log2
3 × 2
52

3. 4. log2 5 = 𝑥
log2 7 = 𝑦
log2 140 = ?
log2 140 = log2 5 × 22 × 7
= log2 5 + 2 log2 2 + log2 7
= 𝑥 + 2 + 𝑦

4. 5. log 𝑥 3 = y
log 𝑥
27
𝑥2 = ?
log 𝑥
27
𝑥2
= log 𝑥
33
𝑥2
= 3 log 𝑥 3 − 2 log 𝑥 𝑥
= 3𝑦 − 2

5. 6. 𝑥 = 7 𝑝
y = 7 𝑞
log7
343𝑥2
𝑦2 = ?
log7
343𝑥2
𝑦2
= log7
73 𝑥2
𝑦2
= 3 log7 7 + 2 log7 𝑥 −2 log7 𝑦
= 3 + 2𝑝 − 2𝑞
= 3 +2 log7 7 𝑝 −2 log7 7 𝑞

6. log27 𝑎 = log27 3 𝑝
= p log27 3
7. 𝑎 = 3 𝑝
log27 𝑎 = ?
a𝑠𝑠𝑢𝑚𝑒 log27 3 = 𝑦
27 𝑦 = 3
33 𝑦 = 31
3𝑦 = 1
𝑦 =
1
3
∴ log27 𝑎 =
𝑝
3
8. log 𝑎 𝑥 = 𝑚
log1
𝑎
𝑥 = ?
𝑎 𝑚 = 𝑥
log1
𝑎
𝑥 = log1
𝑎
𝑎 𝑚
a𝑠𝑠𝑢𝑚𝑒 log1
𝑎
𝑎 𝑚 = 𝑦
1
𝑎
𝑦
= 𝑎 𝑚
𝑎−𝑦 = 𝑎 𝑚
𝑦 = −𝑚

7. 11. 𝑓. log3
27
5
+ log3
10
3
− log3
2
3
= 3 log3 3 − log3 5 + log3 10 − log3 3 − log3 2 + log3 3
= 3
= 3 − log3 5 + log3 10 − log3 2
= 3 − log3 5 + log3 5 × 2 − log3 2
= 3 − log3 5 + log3 5 + log3 2 − log3 2

8. 12. log 2 = a
log 3 = 𝑏
𝑐. log 3 = log 3
1
2
=
1
2
log 3
=
𝑏
2
ℎ. log 0.75 = log
3
4
= log 3 − 2 log 2
= 𝑏 − 2𝑎