If $p^3=q^4=r^6=t^7=s^2$, then $\log _t(p q r s)=\ldots$.

Let $p^3 = q^4 = r^6 = t^7 = s^2 = k$. Then $p = k^{1/3}$, $q = k^{1/4}$, $r = k^{1/6}$, $t = k^{1/7}$, and $s = k^{1/2}$.

The product $pqrs = k^{1/3} \cdot k^{1/4} \cdot k^{1/6} \cdot k^{1/2} = k^{(1/3) + (1/4) + (1/6) + (1/2)}$. Converting to a common denominator of 12 yields $\frac{4}{12} + \frac{3}{12} + \frac{2}{12} + \frac{6}{12} = \frac{15}{12} = \frac{5}{4}$, so $pqrs = k^{5/4}$.

Evaluating $\log_t(pqrs)$ gives $\log_{k^{1/7}}(k^{5/4}) = \frac{5/4}{1/7} \log_k k = \frac{5}{4} \cdot 7 = \frac{35}{4}$.

Final answer: $\frac{35}{4}$.