Logarithmic Relationship Analysis

The problem asks to identify the true relationship among P, Q, and R given the logarithmic equality:

Deriving Variable Relationships

We use the properties of logarithms to establish connections between the variables. Let the common value of the expressions be $k$. This means:

• $\log(P) = k \implies P = b^k$ (where $b$ is the base of the logarithm)
• $\frac{1}{2}\log(Q) = k \implies \log(Q) = 2k \implies Q = b^{2k}$
• $\frac{1}{3}\log(R) = k \implies \log(R) = 3k \implies R = b^{3k}$

Alternatively, using logarithm properties directly:

• From $\log(P) = \frac{1}{2}\log(Q)$, we get $2\log(P) = \log(Q)$, which implies $\log(P^2) = \log(Q)$, so $P^2 = Q$.
• From $\log(P) = \frac{1}{3}\log(R)$, we get $3\log(P) = \log(R)$, which implies $\log(P^3) = \log(R)$, so $P^3 = R$.

We now have $Q = P^2$ and $R = P^3$. We will use these to check the given options.

Evaluating Options

Substitute $Q = P^2$ and $R = P^3$ into each option:

1. Option 1: $P^2 = Q^3R^2$
   $ P^2 = (P^2)^3 (P^3)^2 $
   $ P^2 = P^6 \cdot P^6 $
   $ P^2 = P^{12} $

   This is not generally true.

2. Option 2: $Q^2 = PR$
   $ (P^2)^2 = P \cdot (P^3) $
   $ P^4 = P^4 $

   This statement is TRUE.

3. Option 3: $Q^2 = R^3P$
   $ (P^2)^2 = (P^3)^3 \cdot P $
   $ P^4 = P^9 \cdot P $
   $ P^4 = P^{10} $

   This is not generally true.

4. Option 4: $R = P^2Q^2$
   $ P^3 = P^2 \cdot (P^2)^2 $
   $ P^3 = P^2 \cdot P^4 $
   $ P^3 = P^6 $

   This is not generally true.

Conclusion

The relationship $Q^2 = PR$ holds true based on the initial logarithmic conditions.