Common Logarithms Questions and Answers - Form 2 Topical Mathematics

Questions

1. Use mathematical table to evaluate.
   
   
2. Given that y = Bxⁿ. Make n the subject of the formula and simplify your answer
3. Without using mathematical tables or calculators evaluate: 6log₂64 + 10log₃(243)
4. Find the value of x that satisfies the equation log (2x – 11) – log 2 =log 3 – log x
5. Use logarithms to evaluate to 3 significant figures
   
   
6. Use logarithm tables in all your steps to evaluate: 
   
   leaving your answer to four decimal places
7. Make L the subject in :
   
   
8. Using logarithm tables solve.
   
   
9. Solve the simultaneous equation:-
   Log (x-1) + 2log y = 2log3
   log x + log y = log 6
10. Without using logarithms tables or calculator evaluate:-
    ⁴/₅log₁₀32 + log₁₀50-3log₁₀
11. Use logarithms to evaluate:-
    
     and express the answer in standard form
12. Solve for x given that :- log (3x + 8) – 3log2 = log (x-4)
13. 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:
14. Use logarithms to evaluate correct to 4 s.f
    
    
15. Without using logarithm tables evaluate:
    
    
16. Without using a calculator/mathematical tables, solve: Log₈ (x + 5) – log₈(x -3) = Log₈4
17. Use tables to calculate;(6.57² + 6.57) ÷ (7.92² x 30.08)(Give your answer to 4 decimal places)
18. If log2 = 0.30103, and log3 = 0.47712, calculate without using tables or calculators the value of log120
19. Solve for x in the following equation; Log₂(3x -4) = ¹/₃log₂8x⁶ – log₂4
20. By showing all the steps, use logarithms to evaluate: 
    
    
21. Solve the logarithimic equation: log₁₀(6x – 2) – 1 = log₁₀ (x - 3)
22. 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:-
    
    
23. Evaluate using logarithms
    
    

Answers

1.  
   
2. Log y = log B + n log x
   n log x = log y – log B
   n = Log (^y/_B)/_{Log x}
3. = 6 log₂4 + 10 log₃3
   = 12 log₂2 + 10 log₃3
   = 12 + 10
4. Log ^{(2x - 11)}/₂ = ^{log 3}/_x
   (2x – 11) = ³/_x
   2x² _ 11x -6 = 0
   (2x + 1 ) (x – 6) = 0
   x = - ½ or 6
   x = 6
5.  
   
6.  
   
7. H³ = ^{3d(L - d)}/_10L
   3dL - 10H³L= 3d²
   L(3d -10H³)3d²
   L = 3d²/_{3d - 10H³}
8.  
   
9. 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)}/_y = 9
   6y – y² = 9
   y²– 6y + 9 = 0
   (y-3)² = 0
   y = 3
   ∴x = 2
10. ⁴/₅ log₁₀25 + log₁₀25x² – log 10
    4log 2 = log₁₀25x2 – 3log2
    2log10 + 2log5
    Log 10 x 100
11.  
    
12. Log 3x + 8 – log 8 = log (x-4)
    Log ^{(3x + 8})/₈ = log (x-4)
    3x + 8 = x -4
    3x + 8 = 8x – 32
    5x = 40
13.  
    
14.  
    
15. From square roots 12.25 = 3.5
    3.264 x 1.215 x 3.5x√107
    1.088 x 0.4725 x 107
    3264 x 1215 x35
    1088 x 4725
    √27 = 3
16. Log₈(x + 5) – log₈(x -3) = Log₈4
    Log₈^{(x + 5)}/_{x – 3} = log₈4
    x + 5 = 4
    x – 3
    4x – 12 = x + 5
    3x = 17
    x = ¹⁷/₃ = 5²/₃
    Or log₈^x+5/_x-3 = ²/₃
    8 ²/₃ = ^{x + 5}/_{x – 3}
    2³(²/₃) = ^{x + 5}/_{x -3}
    2² = ^{x +5}/_{x - 3} →4 = ^{x + 5}/_{x - 3}
    4x -12 = x + 5  3x = 17
    x = ¹⁷/₃ = 5²/₃
17.  
    
18. Log 120 = log 4 + log 3 + log 10
    = log22 + log3 + log 10
    = 2log2 + log3 + log 10
    = 2(0.30103) + 0.47712 + 1
    = 2.07918
19. Log₂ (3x – 4) = ¹/₃ lo₂8x⁶ – log₂4
    Log₂ (3x – 4) = log₂(23x⁶) - log₂4
    Log₂ (3x – 4) = log₂2x² – 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
20.  
    
21. Det 2 - -3 = 5
    Area of A^IB^IC^I = 5 x 15
    = 75 cm²
22. Log10(6x-2) – log10 = log10(x-3)
    Log (^{6x -2)}/₁₀ = log (x-3)
    ^{6x - 2}/₁₀ = x -3
    6x – 2 = 10x -30
    x = 7
23. No.                    Log
    0.075262           2.8766 x 2 = 3.7532
    6.652                 0.8230 = 0.8230
    ^4.9302/₃ = 6 + ^2.9302/₃
    = 2.9767
    Antilog = 9.4776 x 10^-2
    = 0.094776(accept 0.09478)