Q.5
For positive non-zero real variables p and q, if
log (p2 + q2) = log p + log q + 2 log 3, then the value of
(p4 + q4) / (p2q2) is
and algebraic simplification. Given a logarithmic relation for positive real numbers
p and q, we evaluate the expression
(p⁴ + q⁴)/(p²q²).
Given Equation
log(p² + q²) = log p + log q + 2 log 3
Assume all logarithms are of the same base greater than 1, and
p, q > 0.
Step-by-Step Solution
Step 1: Use Logarithm Properties
log p + log q = log(pq) and 2 log 3 = log(3²) = log 9
Therefore:
log(p² + q²) = log(9pq)
Step 2: Remove Logarithms
Since logarithms are equal:
p² + q² = 9pq
Step 3: Divide by pq
(p²)/(pq) + (q²)/(pq) = 9
p/q + q/p = 9
Step 4: Square the Expression
(p/q + q/p)² = 81
Expanding:
p²/q² + 2 + q²/p² = 81
p²/q² + q²/p² = 79
Step 5: Evaluate the Required Expression
(p⁴ + q⁴)/(p²q²) =
(p²/q²) + (q²/p²)
= 79
Alternate Method
(p⁴ + q⁴)/(p²q²) =
[(p² + q²)² − 2p²q²] / (p²q²)
Substitute p² + q² = 9pq:
(81p²q²)/(p²q²) − 2 = 81 − 2 = 79
Options Analysis
| Option | Value | Correct? | Reason |
|---|---|---|---|
| (A) | 79 | ✅ Yes | Matches p²/q² + q²/p² derived from expansion |
| (B) | 81 | ❌ No | Equals (p/q + q/p)², ignores −2 term |
| (C) | 9 | ❌ No | Only equals p/q + q/p |
| (D) | 83 | ❌ No | No algebraic justification |
Key Insights
- Convert logarithmic equations to algebraic form early
- Use identities like (a + b)² = a² + 2ab + b²
- Avoid solving quadratic unnecessarily when symmetry helps
Final Answer
The correct answer is (A): 79
