Common Logarithms Questions and Answers

Common Logarithms Questions

1. The product of a and is 31.59. Given that logarithm of a is 2.6182. Find using logarithm the value of b. to 4 significant figures. (4mks)

2. Evaluate without using mathematical tables or calculators,

2 log₁₀ 5 – ½ log₁₀ 64 + 2 log₁₀ 40. (3mks)

3. Use logarithm table to evaluate. (4 marks)

3 (0.0246)² x 142

0.002 x 1.14

4. Without using log tables or a calculator; solve (4mks)

5. Solve for x given

 1 ^x
.64² = 256 (3 marks)

 8

6. Use logarithms to evaluate (4mks)

7. Use logarithms to evaluate (4 marks)

8. Use mathematical table to evaluate.

2849 x 0.00574

36.89 ÷ 0.023

9. Given that y = Bxⁿ. Make n the subject of the formula and simplify your answer

10. Without using mathematical tables or calculators evaluate: 6log₂ 64 + 10log₃ (243)

11. Find the value of x that satisfies the equation log (2x – 11) – log 2 =log 3 – log x

12. Use logarithms to evaluate to 3 significant figures

(0.5241)² x 83.59

³ 0.3563

13. Use logarithm tables in all your steps to evaluate:

leaving your answer to four decimal places

14. Make L the subject in :

H =

15. Using logarithm tables solve.

6.195 x 11.82

83.52

16. Solve the simultaneous equation:-

 Log (x-1) + 2log y = 2log3

log x + log y = log 6

17. Without using logarithms tables or calculator evaluate:-

4 log₁₀32 + log₁₀50-3log₁₀ 2

5

18. Use logarithms to evaluate:-

6.598

(0.9895)² x 0.004974 ^0.75 and express the answer in standard form

19. Solve for x given that :- log (3x + 8) – 3log2 = log (x-4)

20. In this question, show all the steps in your calculations, giving your answer at each stage.

Use logarithms correct to 4 decimal places to evaluate:

36.72 x (0.46)²

185.4

21. Use logarithms to evaluate correct to 4 s.f

sin 44.5

tan 14.90 x cos 82

22. Without using logarithm tables evaluate:

3.264 x 1.215 x 12.25

1.088 x 0.4725

23. Without using a calculator/mathematical tables, solve:
Log₈ (x + 5) – log₈(x -3) = Log₈ 4

24. Use tables to calculate (6.57² + 6.57) ÷ (7.92² x 30.08)(Give your answer to 4 decimal places)

25. If log^₂ = 0.30103, and log^₃ = 0.47712, calculate without using tables or calculators the

value of log120

26. Solve for x in the following equation; Log₂(3x -4) = 1 log₂8x⁶ – log₂4 3

27. By showing all the steps, use logarithms to evaluate: 5.627 x (0.234)³

 (8.237) ^½

28. Solve the logarithimic equation: log₁₀ (6x – 2) – 1 = log₁₀ (x-3)

29. In this question, show all the steps in your calculations, giving your answers at each stage.

Use logarithms, correct to 4 d.p to evaluate:-

(0.07526)²

1.789 + 4.863

30. Evaluate using logarithms

Common logarithms Answers

5. 1 ^x
. 2^6
2 = 2^{4 2} M1 for writing in

2³ index form

2^-3x.2¹² = 2⁸

12 – 3x = 8 M1

x = ⁴/₃

= 1 ¹/₃

 3

8. No. Log

2849 3.4547

+

0.00574 3. 7589

1.2136

36.891.1 1. 5669

0.023 2.3617

3.2052

2. 0084 x ¼

3.178 x 10^-1 1. 5021

0.3178

9. Log y = log B + n log x

n log x = log y – log B

n = Log (^y/_B)

Log x

10. = 6 log₂ 4 + 10 log₃3

 = 12 log₂2 + 10 log₃3

 = 12 + 10

11. Log 2x – 11 = log 3

2 x

(2x – 11) = ³/_x

2x² _ 11x -6 = 0

(2x + 1 ) (x – 6) = 0

x = – ½ or 6

x = 6

12.

13. No. Log

38.32 1.5834

12.964 1.1127

2.6961

86.37 1.9364

6.285 0.7783

2.7347

1.9587

-3 + 2.9587 = 1.9866

3 = 0.9695

14. H³ = 3d(L-d)

10L

3dL -10H³L= 3d²

L(3d -10H³) 3d²

 L = 3d²

3d -10H³

15. No. Log

6.195 0.7920

11.82 1.0726

1.8646

 83.52 1.9218

1.9428 x ¼

4. + 3.9428

4

0.9676 1.9857

16. Log y² (x-1) = log 9 y² (x-1) = 9 ….(1)

log (xy) log 6 xy = 6 ….2

from (2) x = ⁶/_y

substitute in (1) y(6 -1) = 9

y

6y – y² = 9

y² – 6y + 9 = 0

(y-3)² = 0

y = 3

x = 2

5

17. ⁴/₅ log₁₀25 + log₁₀ 25×2 – log 10

4 log 2 = log ₁₀ 25×2 – 3log 2

2 log 10 + 2 log 5

Log 10 x 100

18.

19. Log 3x + 8 – log 8 = log (x-4)

Log (3x + 8) = log (x-4)

 8

3x + 8 = x -4

3x + 8 = 8x – 32

5x = 40

20.

21. No Log

Sin 44.5 1.8457

Tan 14.9 1.4250 2.5686 –

Cos 82 1.1486 +

1.2772

2

10 x 4.351 __________ 0.6386

22. From square roots 12.25 = 3.5

3.264 x 1.215 x 3.5 x 107

1.088 x 0.4725 x 107

3264 x 1215 x35

1088 x 4725

27 = 3

23. Log₈ (x + 5) – log₈(x -3) = Log₈ 4

Log₈ (x + 5) = log₈ 4

x – 3

x + 5 = 4

x – 3

4x – 12 = x + 5

3x = 17

x = 17 = 5²/₃

Or log ₈ x +5 = 2

x – 3 3

8 ²/₃ = x + 5

x – 3

2³( ²/₃) = x + 5

x -3

2² = x +5
 4 = x + 5

x -3 x – 3

4x -12 = x + 5  3x = 17

x = 17 = 5²/₃

3

24.

25. Log 120 = log 4 + log 3 + log 10

= log22 + log3 + log 10

= 2log2 + log3 + log 10

= 2(0.30103) + 0.47712 + 1

= 2.07918

26. Log₂ (3x – 4) = ¹/₃ lo₂ 8x⁶ – log₂ 4

 Log₂ (3x – 4) = log₂ (2³x⁶) – log₂ 4

Log₂ (3x – 4) = log₂ 2x² – log₂ 4

 Log₂ (3x – 4) – log₂
2x²

4

 = 3x – 4 = 2x²

4

2x² – 12x + 16 = 0

x² – 6x + 8 = 0

x – 2x – 4x + 8 = 0

(x – 2) (x- 4) = 0

 x = 2 or x = 4

27.

28. Det 2 – -3 = 5

Area of A^IB^IC^I = 5 x 15

= 75 cm²

29. Log10(6x-2) – log10 = log10(x-3)

Log (6x -2) = log (x-3)

10

6x -2 = x -3

10

6x – 2 = 10x -30

x = 7

30. No. Log

0.07526² 2.8766 x 2 = 3.7532

6.652 0.8230 = 0.8230

4.9302

4.9302 = 6 + 2.9302

3 3

 = 2.9767

Antilog = 9.4776 x 10^-2

= 0.094776(accept 0.09478)

No. Log

1. 0.6317
   

0.009478² 3.9767 X 2 +

5. 9534

4.5851 –

Log 9.814 1.9964

4. 5887
 5

2.0785 X 10^-1 1.3177

= 0.20785