GATE CS 2011 Q57 — General Aptitude

Let the given logarithmic relation be equal to a constant $k$:

Using the property of logarithms $a \log(x) = \log(x^a)$, we can rewrite the terms as:

Taking the antilogarithm on all sides, we get:

Now, we express $Q$ and $R$ in terms of $P$:

• $Q = P^2$
• $R = P^3$

Let's verify the given options using these relations:

• Option A: $P^2 = Q^3 R^2 \implies P^2 = (P^2)^3 (P^3)^2 = P^{12}$ (False)
• Option B: $Q^2 = PR \implies (P^2)^2 = P \cdot P^3 \implies P^4 = P^4$ (True)
• Option C: $Q^2 = R^3 P \implies (P^2)^2 = (P^3)^3 \cdot P \implies P^4 = P^{10}$ (False)
• Option D: $R = P^2 Q^2 \implies P^3 = P^2 (P^2)^2 = P^6$ (False)

Thus, the correct option is B.