Question

If $p^{3}=q^{4}=r^{6}=t^{7}=s^{2}$, then $\log_{t}(pqrs)=......$

• A. $168/5$
• B. $28$
• C. $31/4$
• D. $35/4$

Hint:

When variables are equal to each other, raising them to fractional powers of a single base is the easiest substitution.

Answer 1

The Correct Option is D

Step 1: Concept
Express all variables in terms of $t$.

Step 2: Analysis
$p = t^{7/3}$, $q = t^{7/4}$, $r = t^{7/6}$, $s = t^{7/2}$.

Step 3: Calculation
$pqrs = t^{7/3} \cdot t^{7/4} \cdot t^{7/6} \cdot t^{7/2}$
$= t^{7(1/3 + 1/4 + 1/6 + 1/2)} = t^{7(\frac{4+3+2+6}{12})} = t^{7(\frac{15}{12})} = t^{7(5/4)} = t^{35/4}$.
$\log_t(t^{35/4}) = 35/4$.

Step 4: Conclusion
Hence, the value is $35/4$. Final Answer: (D)