Question

If p3 = q 4 = r5 = s6 , then the value of logs(pqr) is equal to

• A. 16/5
• B. 1
• C. 24/5
• D. 47/10

Answer 1

The Correct Option is D

Step 1: Let Each Variable Be a Root of \( k \) 

We are given:

\[ p^3 = q^4 = r^5 = s^6 = k \Rightarrow p = k^{1/3}, \quad q = k^{1/4}, \quad r = k^{1/5}, \quad s = k^{1/6} \]

Step 2: Evaluate the Product \( pqr \)

\[ pqr = k^{1/3} \cdot k^{1/4} \cdot k^{1/5} = k^{\left(\frac{1}{3} + \frac{1}{4} + \frac{1}{5}\right)} \]

Take LCM of denominators: LCM of 3, 4, and 5 is 60

\[ \frac{1}{3} + \frac{1}{4} + \frac{1}{5} = \frac{20 + 15 + 12}{60} = \frac{47}{60} \Rightarrow pqr = k^{47/60} \]

Step 3: Evaluate \( \log_k (pqr) \)

Use the logarithmic identity: \( \log_b (a^n) = n \log_b a \)

\[ \log_k(pqr) = \log_k(k^{47/60}) = \frac{47}{60} \cdot \log_k k = \frac{47}{60} \cdot 1 = \frac{47}{60} \]

Step 4: Express in Terms of Base \( k^{1/6} \)

Now express the log in base \( k^{1/6} \):

\[ \log_{k^{1/6}}(pqr) = \frac{\log_k(pqr)}{\log_k(k^{1/6})} = \frac{\frac{47}{60}}{\frac{1}{6}} = \frac{47}{60} \cdot 6 = \boxed{\frac{47}{10}} \]

Final Answer:

\[ \boxed{\frac{47}{10}} \]