Dear student We have,13log3M+3log3N=1+log0 0085⇒13log3M13N=1+log10-log2log8-log1000⇒13log3M13N=1-1-log231-log2⇒13log3M13N=1-13⇒13log3M13N=23log3M13=323MN9=32N9=9M Regards

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W e h a v e, \frac{1}{3} \log_{3} M + 3 \log_{3} N = 1 + \log_{0.008} 5 \Rightarrow \frac{1}{3} \log_{3} (\frac{M^{\frac{1}{3}}}{N}) = 1 + \frac{\log (10) - \log (2)}{\log (8) - \log (1000)} \Rightarrow \frac{1}{3} \log_{3} (\frac{M^{\frac{1}{3}}}{N}) = 1 - \frac{1 - \log (2)}{3 (1 - \log (2))} \Rightarrow \frac{1}{3} \log_{3} (\frac{M^{\frac{1}{3}}}{N}) = 1 - \frac{1}{3} \Rightarrow \frac{1}{3} \log_{3} (\frac{M^{\frac{1}{3}}}{N}) = \frac{2}{3} \log_{3} (M^{\frac{1}{3}}) = 3^{\frac{2}{3}} M N^{9} = 3^{2} N^{9} = \frac{9}{M}

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