Explanation of logarithmic functions with solved examples.

1. Logarithms
Prepared by
Ismail Mohammad El-Badawy
ismailelbadawy@gmail.com

2. Logarithmic and exponential forms
• Log 2 8
• Log 3 81
• Log 6 36
2 to the power of ?? gives 8
3 to the power of ?? gives 81
6 to the power of ?? gives 36
What is the base 2 logarithm of 8 ?
What is the base 3 logarithm of 81?
What is the base 6 logarithm of 36?

3. Logarithmic and exponential forms
• Log 2 8 = 3 23 = 8
• Log 3 81 = 4 34 = 81
• Log 6 36 = 2 62 = 36
2 to the power of 3 gives 8
3 to the power of 4 gives 81
6 to the power of 2 gives 36
What is the base 2 logarithm of 8 ? 3
What is the base 3 logarithm of 81? 4
What is the base 6 logarithm of 36? 2

4. Logarithmic and exponential forms
Note: The default value of
the base is 10.
e.g. Log (100) = 2
102 = 100

5. • Find the value of a :
• Log 4 𝑎 = 3
• Log 2 𝑎 = 5
• Log 5 𝑎 = 4
The argument is unknown

6. • Find the value of a :
• Log 4 𝑎 = 3
✓ 43
= 𝑎 ∴ 𝑎 = 64
• Log 2 𝑎 = 5
✓ 25
= 𝑎 ∴ 𝑎 = 32
• Log 5 𝑎 = 4
✓ 54 = 𝑎 ∴ 𝑎 = 625
The argument is unknown
Check your answers
𝐿𝑜𝑔4 64 =
𝐿𝑜𝑔 64
𝐿𝑜𝑔 4
𝐿𝑜𝑔2 32 =
𝐿𝑜𝑔 32
𝐿𝑜𝑔 2
𝐿𝑜𝑔5 625 =
𝐿𝑜𝑔 625
𝐿𝑜𝑔 5

7. • Find the value of b :
• Log 𝑏 81 = 2
• Log 𝑏 243 = 5
• Log 𝑏 216 = 3
The base is unknown

8. • Find the value of b :
• Log 𝑏 81 = 2
✓ 𝑏2
= 81 → 𝑏2
= 92
∴ 𝑏 = 9
• Log 𝑏 243 = 5
✓ 𝑏5
= 243 → 𝑏5
= 35
∴ 𝑏 = 3
• Log 𝑏 216 = 3
✓ 𝑏3 = 216 → 𝑏3 = 63 ∴ 𝑏 = 6
The base is unknown
Check your answers
𝐿𝑜𝑔9 81 =
𝐿𝑜𝑔 81
𝐿𝑜𝑔 9
𝐿𝑜𝑔3 243 =
𝐿𝑜𝑔 243
𝐿𝑜𝑔 3
𝐿𝑜𝑔6 216 =
𝐿𝑜𝑔 216
𝐿𝑜𝑔 6

9. • Find the value of c :
• Log 3 81 = 𝑐
• Log 7 49 = 𝑐
• Log 5 125 = 𝑐
The exponent is unknown

10. • Find the value of c :
• Log 3 81 = 𝑐
✓ 3 𝑐
= 81 → 3 𝑐
= 34
∴ 𝑐 = 4
• Log 7 49 = 𝑐
✓ 7 𝑐 = 49 → 7 𝑐 = 72 ∴ 𝑐 = 2
• Log 5 125 = 𝑐
✓ 5 𝑐 = 125 → 5 𝑐 = 53 ∴ 𝑐 = 3
The exponent is unknown
Check your answers
𝐿𝑜𝑔3 81 =
𝐿𝑜𝑔 81
𝐿𝑜𝑔 3
𝐿𝑜𝑔7 49 =
𝐿𝑜𝑔 49
𝐿𝑜𝑔 7
𝐿𝑜𝑔5 125 =
𝐿𝑜𝑔 125
𝐿𝑜𝑔 5

11. • Find the value of x :
• Log 7 (2𝑥 − 5) = 3
• Log (𝑥2) = 4
• Log 5 3𝑥 = 1.6

12. • Find the value of x :
• Log 7 (2𝑥 − 5) = 3
✓ 73
= 2𝑥 − 5 → 343 = 2𝑥 − 5 ∴ 𝑥 = 343 + 5 ÷ 2 = 174
• Log (𝑥2) = 4
✓ 104
= 𝑥2
→ 10000 = 𝑥2
∴ 𝑥 = ± 10000 = ±100
• Log 5 3𝑥 = 1.6
✓ 51.6
= 3𝑥 → 13.1 = 3𝑥 ∴ 𝑥 = 13.1 ÷ 3 = 4.37

13. • Find the value of x :
• Log (𝑥−3) 5 = 0.5
• Log 2 16 = 7𝑥 − 3

14. • Find the value of x :
• Log (𝑥−3) 5 = 0.5
✓ (𝑥 − 3)0.5
= 5 → 𝑥 − 3 = 5
→ 𝑥 − 3 = 52
→ 𝑥 − 3 = 25 ∴ 𝑥 = 28
• Log 2 16 = 7𝑥 − 3
✓ 2(7𝑥−3)
= 16 → 2(7𝑥−3)
= 24
→ 7𝑥 − 3 = 4 ∴ 𝑥 = 1

15. • Log 4 4 = ? ?
• Log 3 1 = ? ?
• Log 6 ? ? = 1
• Log 5 ? ? = 0
• Log 2 12 = Log 2 3 + Log 2 ? ?
• Log 𝑥 (? ? ) = Log 𝑥 5 + Log 𝑥 𝑦
Rules of logs

16. • Log 4 4 = 1
• Log 3 1 = 0
• Log 6 6 = 1
• Log 5 1 = 0
• Log 2 12 = Log 2 3 + Log 2 4
• Log 𝑥 (5𝑦) = Log 𝑥 5 + Log 𝑥 𝑦
Rules of logs

17. • Log 3 25 = ? ? Log 3 5
• 4 Log 7 2 = Log 7 (? ? )
• Log 2 12 = Log 2 36 − Log 2 ? ?
• Log
??
??
= Log 𝑥 − Log 5𝑦
• Log 4
2
10
= −Log 4 (? ? )
• Log 3 25 =
Log 𝑥 (??)
Log 𝑥 (??)
Rules of logs

18. • Log 3 25 = 2 Log 3 5
• 4 Log 7 2 = Log 7 (16)
• Log 2 12 = Log 2 36 − Log 2 3
• Log
𝑥
5𝑦
= Log 𝑥 − Log 5𝑦
• Log 4
2
10
= −Log 4 (5)
• Log 3 25 =
Log 𝑥 (25)
Log 𝑥 (3)
Rules of logs

19. • Evaluate:
• Log 2
1
16
• Log 16 8 =
• Log 5 1
Using rules of Logs

20. • Evaluate:
• Log 2
1
16
✓ Log 2
1
16
= −Log 2 16 = −4
• Log 16 8
✓ Log 16 8 =
Log 2 8
Log 2 16
=
3
4
• Log 5 1 = 0
Using rules of Logs

21. • Evaluate:
• Log 5 3 − Log 5 75
• 2 Log 2 6 − Log 2 9
•
1
2
Log 10 4 + Log 10 50
Using rules of Logs

22. • Evaluate:
• Log 5 3 − Log 5 75
✓ Log 5
3
75
= Log 5
1
25
= −Log 5 25 = −2
• 2 Log 2 6 − Log 2 9
✓ Log 2 62 − Log 2 9 = Log 2 36 − Log 2 9 = Log 2
36
9
= Log 2 4 = 2
•
1
2
Log 10 4 + Log 10 50
✓ Log 10 40.5 + Log 10 50 = Log 10 2 + Log 10 50 = Log 10 2 × 50 = Log 10 100 = 2
Using rules of Logs

23. • Simplify the following logarithms:
• Log 10 + Log 2 − Log 4
•
1
2
Log 𝑥 − 3 Log y
•
1
3
Log 8 + 2 Log 4
Using rules of Logs

24. • Simplify the following logarithms:
• Log 10 + Log 2 − Log 4
✓ Log 10 × 2 − Log 4 = Log 20 − Log 4 = Log
20
4
= Log 5
•
1
2
Log 𝑥 − 3 Log y
✓ Log 𝑥 Τ1
2 − Log 𝑦3 = Log
𝑥
𝑦3
•
1
3
Log 8 + 2 Log 4 = Log 8 Τ1
3 + Log 42 = Log 2 + Log 16 = Log 2 × 16 = Log 32
Using rules of Logs

25. • The point P on the curve 𝑦 = 9 𝑥 has y-coordinate equal to 150. Use logarithms to find the
x-coordinate of P, correct to 3 significant figures.
Using rules of Logs

26. • The point P on the curve 𝑦 = 9 𝑥 has y-coordinate equal to 150. Use logarithms to find the
x-coordinate of P, correct to 3 significant figures.
✓ 150 = 9 𝑥 Log 150 = Log 9 𝑥
Log 150 = 𝑥 Log 9
∴ 𝑥 =
Log 150
Log 9
= Log9150 = 2.28
Using rules of Logs

27. • Given that Log 𝑥 5𝑦 + 1 − Log 𝑥 3 = 4, express y in terms of x.
Using rules of Logs

28. • Given that Log 𝑥 5𝑦 + 1 − Log 𝑥 3 = 4, express y in terms of x.
✓ Log 𝑥
5𝑦+1
3
= 4
𝑥4
=
5𝑦 + 1
3
3𝑥4
= 5𝑦 + 1
∴ 𝑦 =
3𝑥4 − 1
5
Using rules of Logs

29. • Use logarithms to solve the following equation, giving the value of x correct to 3 s.f:
• 7 𝑥
= 2 𝑥+1
Using rules of Logs

30. • Use logarithms to solve the following equation, giving the value of x correct to 3 s.f:
• 7 𝑥 = 2 𝑥+1
✓ Log 7 𝑥 = Log 2 𝑥+1 → 𝑥 Log 7 = 𝑥 + 1 Log 2
𝑥
𝑥+1
=
Log 2
Log 7
= 0.356
𝑥 = 𝑥 + 1 0.356 = 0.356𝑥 + 0.356
𝑥 − 0.356𝑥 = 0.356
0.644 𝑥 = 0.356
∴ 𝑥 = 0.356 ÷ 0.644 = 0.553
Using rules of Logs

31. Assignment