[tex] \frac{ ^{3}\log_{45} - \: {}^{3} \log_{5} }{ {}^{ \sqrt{3} }\log_{100} \log_{9} - \: {}^{5} \log_{ \sqrt[3]{25} }} \\ \\ \to \frac{ {}^{3}\log_{9} }{ 4 \: . \: {}^{3} \log_{10} (2 \: . \: \log_{3}) - \frac{2}{3} \: . \: {}^{5} \log_{5} } \\ \\ \to \frac{2}{8 - \frac{2}{3} } \\ \\ \to \frac{2}{ \frac{22}{3} } \\ \\ \to \frac{1}{ \frac{11}{3} } \\ \\ \to \green{ \frac{3}{11} }[/tex]
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[tex] \frac{ ^{3}\log_{45} - \: {}^{3} \log_{5} }{ {}^{ \sqrt{3} }\log_{100} \log_{9} - \: {}^{5} \log_{ \sqrt[3]{25} }} \\ \\ \to \frac{ {}^{3}\log_{9} }{ 4 \: . \: {}^{3} \log_{10} (2 \: . \: \log_{3}) - \frac{2}{3} \: . \: {}^{5} \log_{5} } \\ \\ \to \frac{2}{8 - \frac{2}{3} } \\ \\ \to \frac{2}{ \frac{22}{3} } \\ \\ \to \frac{1}{ \frac{11}{3} } \\ \\ \to \green{ \frac{3}{11} }[/tex]