Step 1: Understanding the Question:
We need to express $p, q, r, s$ in terms of $t$ to evaluate the logarithm.
Step 2: Key Formula or Approach:
Property: If $x^a = y^b$, then $x = y^{\frac{b}{a}}$.
Property: $\log_{x^n}(x^m) = \frac{m}{n}$.
Step 3: Detailed Explanation:
Let $p^3 = q^4 = r^6 = t^7 = s^2 = k$.
Then:
$p = k^{\frac{1}{3}}$, $q = k^{\frac{1}{4}}$, $r = k^{\frac{1}{6}}$, $t = k^{\frac{1}{7}}$, $s = k^{\frac{1}{2}}$.
Now calculate the product $pqrs$:
\[ pqrs = k^{\frac{1}{3} + \frac{1}{4} + \frac{1}{6} + \frac{1}{2}} \]
The LCM of denominators $(3, 4, 6, 2)$ is $12$:
\[ pqrs = k^{\frac{4 + 3 + 2 + 6}{12}} = k^{\frac{15}{12}} = k^{\frac{5}{4}} \]
We need $\log_t(pqrs)$:
\[ \log_{k^{1/7}}(k^{5/4}) = \frac{5/4}{1/7} = \frac{5}{4} \times 7 = \frac{35}{4} \]
Step 4: Final Answer:
The value is $\frac{35}{4}$.