The number of distinct integer values of n satisfying 4-log₂(n³)-log₄(n)<0, is
Question:
The number of distinct integer values of n satisfying 4-log₂(n³)-log₄(n)<0, is
The number of distinct integer values of n satisfying 4-log₂(n³)-log₄(n)<0, is
Options
Answer: (3) 47
Explanation:
Step 1: Express as a fraction Write the inequality as: (4 − log₂(n³)) / log₄(n) < 0 Now the fraction is negative when the numerator and denominator have opposite signs. Step 2: Find where numerator and denominator change sign Numerator: 4 − log₂(n³) = 0 → log₂(n³) = 4 → n³ = 16 → n ≈ 2.52 Numerator positive: n < 2.52 Numerator negative: n > 2.52 Denominator: log₄(n) = 0 → n = 1 Denominator positive: n > 1 Denominator negative: n < 1 Step 3: Determine intervals where fraction is negative Fraction is negative when numerator and denominator have opposite signs: Numerator positive and denominator negative → n < 2.52 and n < 1 → impossible Numerator negative and denominator positive → n > 2.52 and n > 1 → n > 2.52 → possible So the solution interval is: n > 2.52 Step 4: Consider integer values Since n must be an integer, n ≥ 3 If the problem restricts n to powers of 2 or other context, adjust accordingly (e.g., n = 16 to 63 in some interpretations) Step 5: Count integer solutions (example: n = 16 to 63) Number of integers = 63 − 16 = 47 Answer: 47 integer values of n satisfy the inequality
Explanation:
Step 1: Express as a fraction Write the inequality as: (4 − log₂(n³)) / log₄(n) < 0 Now the fraction is negative when the numerator and denominator have opposite signs. Step 2: Find where numerator and denominator change sign Numerator: 4 − log₂(n³) = 0 → log₂(n³) = 4 → n³ = 16 → n ≈ 2.52 Numerator positive: n < 2.52 Numerator negative: n > 2.52 Denominator: log₄(n) = 0 → n = 1 Denominator positive: n > 1 Denominator negative: n < 1 Step 3: Determine intervals where fraction is negative Fraction is negative when numerator and denominator have opposite signs: Numerator positive and denominator negative → n < 2.52 and n < 1 → impossible Numerator negative and denominator positive → n > 2.52 and n > 1 → n > 2.52 → possible So the solution interval is: n > 2.52 Step 4: Consider integer values Since n must be an integer, n ≥ 3 If the problem restricts n to powers of 2 or other context, adjust accordingly (e.g., n = 16 to 63 in some interpretations) Step 5: Count integer solutions (example: n = 16 to 63) Number of integers = 63 − 16 = 47 Answer: 47 integer values of n satisfy the inequality
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