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- Question
If ax = by, then:
Options- A.
log a = x b y - B.
log a = x log b y - C.
log a = y log b x - D. None of these
- Correct Answer
-
log a = y log b x
Explanationax = by⟹ log ax = log by
⟹ x log a = y log b
⟹ log a = y . log b x
Logarithm problems
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- 1.
If log a + log b = log (a + b), then: b a
Options- A. a + b = 1
- B. a - b = 1
- C. a = b
- D. a2 - b2 = 1 Discuss
Correct Answer: a + b = 1
Explanation:
log a + log b = log (a + b) b a ⟹ log (a + b) = log ❨ a x b ❩ = log 1. b a So, a + b = 1.
- 2. A boat can travel with a speed of 13 km/hr in still water. If the speed of the stream is 4 km/hr, find the time taken by the boat to go 68 km downstream.
Options- A. 2 hours
- B. 3 hours
- C. 4 hours
- D. 5 hours Discuss
Correct Answer: 4 hours
Explanation:
Speed downstream = (13 + 4) km/hr = 17 km/hr.Time taken to travel 68 km downstream = ❨ 68 ❩hrs = 4 hrs. 17 - 3. A boat running upstream takes 8 hours 48 minutes to cover a certain distance, while it takes 4 hours to cover the same distance running downstream. What is the ratio between the speed of the boat and speed of the water current respectively?
Options- A. 2 : 1
- B. 3 : 2
- C. 8 : 3
- D. Cannot be determined
- E. None of these Discuss
Correct Answer: 8 : 3
Explanation:
Let the man's rate upstream be x kmph and that downstream be y kmph.Then, distance covered upstream in 8 hrs 48 min = Distance covered downstream in 4 hrs.
⟹ ❨ x x 8 4 ❩ = (y x 4) 5 ⟹ 44 x =4y 5 ⟹ y = 11 x. 5 ∴ Required ratio = ❨ y + x ❩ : ❨ y - x ❩ 2 2 = ❨ 16x x 1 ❩ : ❨ 6x x 1 ❩ 5 2 5 2 = 8 : 3 5 5 = 8 : 3.
- 4. A man takes twice as long to row a distance against the stream as to row the same distance in favour of the stream. The ratio of the speed of the boat (in still water) and the stream is:
Options- A. 2 : 1
- B. 3 : 1
- C. 3 : 2
- D. 4 : 3 Discuss
Correct Answer: 3 : 1
Explanation:
Let man's rate upstream be x kmph.Then, his rate downstream = 2x kmph.
∴ (Speed in still water) : (Speed of stream) = ❨ 2x + x ❩ : ❨ 2x - x ❩ 2 2 = 3x : x 2 2 = 3 : 1.
- 5. In one hour, a boat goes 11 km/hr along the stream and 5 km/hr against the stream. The speed of the boat in still water (in km/hr) is:
Options- A. 3 km/hr
- B. 5 km/hr
- C. 8 km/hr
- D. 9 km/hr Discuss
Correct Answer: 8 km/hr
Explanation:
Speed in still water = 1 (11 + 5) kmph = 8 kmph. 2 - 6. If log10 2 = 0.3010, then log2 10 is equal to:
Options- A.
699 301 - B.
1000 301 - C. 0.3010
- D. 0.6990 Discuss
Correct Answer:1000 301
Explanation:
log2 10 = 1 = 1 = 10000 = 1000 . log10 2 0.3010 3010 301 - 7. If log10 2 = 0.3010, the value of log10 80 is:
Options- A. 1.6020
- B. 1.9030
- C. 3.9030
- D. None of these Discuss
Correct Answer: 1.9030
Explanation:
log10 80 = log10 (8 x 10) = log10 8 + log10 10 = log10 (23 ) + 1 = 3 log10 2 + 1 = (3 x 0.3010) + 1 = 1.9030. - 8. The value of log2 16 is:
Options- A.
1 8 - B. 4
- C. 8
- D. 16 Discuss
Correct Answer: 4
Explanation:
Let log2 16 = n.Then, 2n = 16 = 24 ⟹ n = 4.
∴ log2 16 = 4.
- 9. If log10 5 + log10 (5x + 1) = log10 (x + 5) + 1, then x is equal to:
Options- A. 1
- B. 3
- C. 5
- D. 10 Discuss
Correct Answer: 3
Explanation:
log10 5 + log10 (5x + 1) = log10 (x + 5) + 1⟹ log10 5 + log10 (5x + 1) = log10 (x + 5) + log10 10
⟹ log10 [5 (5x + 1)] = log10 [10(x + 5)]
⟹ 5(5x + 1) = 10(x + 5)
⟹ 5x + 1 = 2x + 10
⟹ 3x = 9
⟹ x = 3.
- 10. Which of the following statements is not correct?
Options- A. log10 10 = 1
- B. log (2 + 3) = log (2 x 3)
- C. log10 1 = 0
- D. log (1 + 2 + 3) = log 1 + log 2 + log 3 Discuss
Correct Answer: log (2 + 3) = log (2 x 3)
Explanation:
(a) Since loga a = 1, so log10 10 = 1.(b) log (2 + 3) = log 5 and log (2 x 3) = log 6 = log 2 + log 3
∴ log (2 + 3) ≠ log (2 x 3)
(c) Since loga 1 = 0, so log10 1 = 0.
(d) log (1 + 2 + 3) = log 6 = log (1 x 2 x 3) = log 1 + log 2 + log 3.
So, (b) is incorrect.
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