Q.60 If Log(P) = (1/2) Log(Q) = (1/3) Log(R), then which of the following options is TRUE?
Correct Answer: Option (B) Q² = PR
Option (B) Q² = PR is true. Let k = log P = (1/2) log Q = (1/3) log R, so P = 10ᵏ, Q = 10²ᵏ, and R = 10³ᵏ. Substituting yields Q² = (10²ᵏ)² = 10⁴ᵏ and PR = 10ᵏ · 10³ᵏ = 10⁴ᵏ, confirming equality.
Step-by-Step Solution
- Assign common value: log P = k, log Q = 2k, log R = 3k
- Exponentiate base 10: P = 10ᵏ, Q = 10²ᵏ, R = 10³ᵏ
- Test relations algebraically without assuming base
Option Breakdown
- (A) P² = Q³R²: Left: (10ᵏ)² = 10²ᵏ; Right: 10⁶ᵏ · 10⁶ᵏ = 10¹²ᵏ. Unequal.
- (B) Q² = PR: Left: 10⁴ᵏ; Right: 10⁴ᵏ. ✅ Matches.
- (C) Q² = R³P: Left: 10⁴ᵏ; Right: 10⁹ᵏ · 10ᵏ = 10¹⁰ᵏ. No.
- (D) R = P²Q²: Left: 10³ᵏ; Right: 10⁴ᵏ. False.
Detailed Algebraic Proof
Set k = log P. Then log Q = 2k so Q = P²; log R = 3k so R = P³.
Thus Q² = (P²)² = P⁴ and PR = P · P³ = P⁴. Proven!
| Option | Left Side | Right Side | Exponents Equal? |
|---|---|---|---|
| (A) P² | 10²ᵏ | Q³R²: 10¹²ᵏ | ❌ No |
| (B) Q² | 10⁴ᵏ | PR: 10⁴ᵏ | ✅ Yes |
| (C) Q² | 10⁴ᵏ | R³P: 10¹⁰ᵏ | ❌ No |
| (D) R | 10³ᵏ | P²Q²: 10⁴ᵏ | ❌ No |
Exam Tips for Log Equations
- Equalize via common k: Set all logs equal to one variable
- Raise to powers: Match coefficients in exponents
- Verify numerically: Test P=10, Q=100, R=1000 (satisfies B only)
- Practice GATE numerical aptitude: Focus on exponent balancing
Introduction: Crack Log(P) Equation for GATE Success
Log(P) = (1/2)Log(Q) = (1/3)Log(R) from GATE Q.60 tests logarithm manipulation. Q² = PR emerges correct via exponentiation. Master this with breakdowns and shortcuts!
