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$2)a) log_{81}3 = \frac{1}{ log_{3}81} = \frac{1}{ log_{3}{3}^{4}} = \frac{1}{4 log_{3}3} = \frac{1}{4} = 4 \\ b) | \binom{ \sin\theta }{ \cos\theta } \binom{ - \cos\theta }{ \sin\theta } | = \sin\theta \times \sin\theta- \cos\theta\times (-\cos\theta)={sin}^{2}\theta+{cos}^{2}\theta=1 \\c) \frac{1}{ log_{ab}abc } + \frac{1 }{ log_{bc}abc } + \frac{1}{ log_{ac}abc } = 2 \\ log_{abc}ab + log_{abc}bc+ log_{abc}ac = 2 \\ log_{abc}(ab \times bc \times ac) = 2 \\ log_{abc} {(abc)}^{2} = 2 \\ 2 log_{abc}abc = 2 \\ 2(1) = 2 \\ 2 = 2 \\ LHS=RHS\\Hence \: proved \\ e) log_{2}14 - log_{2}7 = log_{2}( \frac{14}{7} ) = log_{2}2 = 1$