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If log 2 = m, log 3 = a and log 7 = n, then the value of log (10!) in terms of m, a and n is
- a)4m + 6a + n + 1
- b)4m + 6a + n + 2
- c)6m + 4a + n + 1
- d)6m + 4a + n + 2
Correct answer is option 'D'. Can you explain this answer?
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Most Upvoted Answer
If log 2 = m, log 3 = a and log 7 = n, then the value of log (10!) in ...
log (10!) = log 10 + log 9 + log 8 + log 7 + ... + log 1
Then, log (10!) = 1 + 2a + 3m + n + a + m + 1 - m + 2m + a + m
⇒ log (10!) = 6m + 4a + n + 2
Then, log (10!) = 1 + 2a + 3m + n + a + m + 1 - m + 2m + a + m
⇒ log (10!) = 6m + 4a + n + 2
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If log 2 = m, log 3 = a and log 7 = n, then the value of log (10!) in ...
Understanding log(10!)
To find log(10!) in terms of m (log 2), a (log 3), and n (log 7), we need to first express 10! in terms of its prime factors.
Prime Factorization of 10!
10! = 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1
Breaking this down into prime factors, we have:
- 10 = 2 × 5
- 9 = 3^2
- 8 = 2^3
- 7 = 7
- 6 = 2 × 3
- 5 = 5
- 4 = 2^2
- 3 = 3
- 2 = 2
- 1 = 1 (not contributing)
Now, combining these factors:
Total count of each prime factor:
- Count of 2:
1 (from 10) + 3 (from 8) + 1 (from 6) + 2 (from 4) + 1 (from 2) = 8
- Count of 3:
2 (from 9) + 1 (from 6) + 1 (from 3) = 4
- Count of 5:
1 (from 10) + 1 (from 5) = 2
- Count of 7:
1 (from 7) = 1
Expression for log(10!)
Now we can express log(10!) as:
log(10!) = log(2^8) + log(3^4) + log(5^2) + log(7^1)
= 8 log 2 + 4 log 3 + 2 log 5 + log 7
Substituting values
Using the given values:
- log 2 = m
- log 3 = a
- log 5 can be derived from log 10 = log(2 × 5) = 1 → log 5 = 1 - log 2 = 1 - m.
Substituting these into the equation:
log(10!) = 8m + 4a + 2(1 - m) + n
= 8m + 4a + 2 - 2m + n
= 6m + 4a + n + 2
Final Result
Thus, the value of log(10!) in terms of m, a, and n is:
Correct answer: option D) 6m + 4a + n + 2.
To find log(10!) in terms of m (log 2), a (log 3), and n (log 7), we need to first express 10! in terms of its prime factors.
Prime Factorization of 10!
10! = 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1
Breaking this down into prime factors, we have:
- 10 = 2 × 5
- 9 = 3^2
- 8 = 2^3
- 7 = 7
- 6 = 2 × 3
- 5 = 5
- 4 = 2^2
- 3 = 3
- 2 = 2
- 1 = 1 (not contributing)
Now, combining these factors:
Total count of each prime factor:
- Count of 2:
1 (from 10) + 3 (from 8) + 1 (from 6) + 2 (from 4) + 1 (from 2) = 8
- Count of 3:
2 (from 9) + 1 (from 6) + 1 (from 3) = 4
- Count of 5:
1 (from 10) + 1 (from 5) = 2
- Count of 7:
1 (from 7) = 1
Expression for log(10!)
Now we can express log(10!) as:
log(10!) = log(2^8) + log(3^4) + log(5^2) + log(7^1)
= 8 log 2 + 4 log 3 + 2 log 5 + log 7
Substituting values
Using the given values:
- log 2 = m
- log 3 = a
- log 5 can be derived from log 10 = log(2 × 5) = 1 → log 5 = 1 - log 2 = 1 - m.
Substituting these into the equation:
log(10!) = 8m + 4a + 2(1 - m) + n
= 8m + 4a + 2 - 2m + n
= 6m + 4a + n + 2
Final Result
Thus, the value of log(10!) in terms of m, a, and n is:
Correct answer: option D) 6m + 4a + n + 2.
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If log 2 = m, log 3 = a and log 7 = n, then the value of log (10!) in terms of m, a and n isa)4m + 6a + n + 1b)4m + 6a + n + 2c)6m + 4a + n + 1d)6m + 4a + n + 2Correct answer is option 'D'. Can you explain this answer?
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If log 2 = m, log 3 = a and log 7 = n, then the value of log (10!) in terms of m, a and n isa)4m + 6a + n + 1b)4m + 6a + n + 2c)6m + 4a + n + 1d)6m + 4a + n + 2Correct answer is option 'D'. Can you explain this answer? for CAT 2024 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about If log 2 = m, log 3 = a and log 7 = n, then the value of log (10!) in terms of m, a and n isa)4m + 6a + n + 1b)4m + 6a + n + 2c)6m + 4a + n + 1d)6m + 4a + n + 2Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If log 2 = m, log 3 = a and log 7 = n, then the value of log (10!) in terms of m, a and n isa)4m + 6a + n + 1b)4m + 6a + n + 2c)6m + 4a + n + 1d)6m + 4a + n + 2Correct answer is option 'D'. Can you explain this answer?.
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