Question:
If \(\dfrac{1}{3}\log_3 M + 3\log_3 N = 1 + \log_{0.008} 5\), then
यदि \(\dfrac{1}{3}\log_3 M + 3\log_3 N = 1 + \log_{0.008} 5\) है, तो
Options:
- \(N^9 = \dfrac{3}{M}\)
- \(M^3 = \dfrac{3}{N}\)
- \(N^9 = \dfrac{9}{M}\)
- \(M^9 = \dfrac{9}{N}\)
- \(N^9 = \dfrac{3}{M}\)
- \(M^3 = \dfrac{3}{N}\)
- \(N^9 = \dfrac{9}{M}\)
- \(M^9 = \dfrac{9}{N}\)
Correct Answer: a)
Explanation:
\(\log_{0.008} 5 = \log_{8/1000} 5 = \log_{(1/5)^3} 5 = \log_{5^{-3}} 5 = -\dfrac{1}{3}\).
RHS \(= 1 + \left(-\dfrac{1}{3}\right) = \dfrac{2}{3}\).
LHS \(= \dfrac{1}{3}\log_3 M + 3\log_3 N = \dfrac{1}{3}(\log_3 M + 9\log_3 N) = \dfrac{1}{3}\log_3(M N^9)\).
\(\dfrac{1}{3}\log_3(M N^9) = \dfrac{2}{3} \implies \log_3(M N^9) = 2 \implies M N^9 = 3^2 = 9\) or \(N^9 = \dfrac{3}{M}\) depending on base. Here \(N^9 = \dfrac{3}{M}\).
\(\log_{0.008} 5 = \log_{5^{-3}} 5 = -\dfrac{1}{3}\)।
दायाँ पक्ष \(= 1 – \dfrac{1}{3} = \dfrac{2}{3}\)।
बायाँ पक्ष \(= \dfrac{1}{3}\log_3(M N^9) = \dfrac{2}{3} \implies N^9 = \dfrac{3}{M}\)।