a) \(A = \ln \left( {\frac{x}{{x - 1}}} \right) + \ln \left( {\frac{{x + 1}}{x}} \right) - \ln \left( {{x^2} - 1} \right) \)
\(= \ln \left( {\frac{x}{{x - 1}}.\frac{{x + 1}}{x}} \right) - \ln \left( {{x^2} - 1} \right) \)
\(= \ln \frac{{x + 1}}{{x - 1}} - \ln \left( {{x^2} - 1} \right)\)
\(= \ln \frac{{x + 1}}{{\left( {x - 1} \right)\left( {{x^2} - 1} \right)}} \)
\(= \ln \frac{{x + 1}}{{{{\left( {x - 1} \right)}^2}\left( {x + 1} \right)}} \)
\(= \ln \frac{1}{{{{\left( {x - 1} \right)}^2}}}\).
b) \(B = 21{\log _3}\sqrt[3]{x} + {\log _3}\left( {9{x^2}} \right) - {\log _3}9 \)
\(= 21{\log _3}{x^{\frac{1}{3}}} + {\log _3}\left( {9{x^2}} \right) - {\log _3}9\)
\(= 21.\frac{1}{3}{\log _3}x + \left[ {{{\log }_3}\left( {9{x^2}} \right) - {{\log }_3}9} \right] \)
\(= 7{\log _3}x + {\log _3}\left( {\frac{{9{x^2}}}{9}} \right)\)
\(= {\log _3}{x^7} + {\log _3}{x^2} \)
\(= \log \left( {{x^7}.{x^2}} \right) \)
\(= {\log _3}{x^9}\).
