31.
$$\frac{1}{2}\left( {\log x + \log y} \right)$$    will equal to $$\log \left( {\frac{{x + y}}{2}} \right)$$   if -

32.
If $$\log \frac{a}{b} + \log \frac{b}{a} = $$   $$\,\log \left( {a + b} \right),$$   then -

33.
$${\log \left( {\frac{{{a^2}}}{{bc}}} \right) + }$$   $${\log \left( {\frac{{{b^2}}}{{ac}}} \right) + }$$   $${\log \left( {\frac{{{c^2}}}{{ab}}} \right)}$$   is equal to -

34.
$$\frac{1}{{{{\log }_a}b}} \times \frac{1}{{{{\log }_b}c}} \times \frac{1}{{{{\log }_c}a}}$$     is equal to -

35.
$${\frac{1}{{\left( {{{\log }_a}bc} \right) + 1}} + }$$   $${\frac{1}{{\left( {{{\log }_b}ca} \right) + 1}} + }$$   $${\frac{1}{{\left( {{{\log }_c}ab} \right) + 1}}}$$   is equal to -