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

32.
The value of $$\left( {\frac{1}{{{{\log }_3}60}} + \frac{1}{{{{\log }_4}60}} + \frac{1}{{{{\log }_5}60}}} \right)$$      is -

33.
If $$\log \frac{a}{b} + \log \frac{b}{a} = $$   $$\,\log \left( {a + b} \right),$$   then -

34.
$$\left[ {\log \left( {\frac{{{a^2}}}{{bc}}} \right) + \log \left( {\frac{{{b^2}}}{{ac}}} \right) + \log \left( {\frac{{{c^2}}}{{ab}}} \right)} \right]$$       is equal to -

35.
$$\frac{1}{{{{\log }_a}b}} \times \frac{1}{{{{\log }_b}c}} \times \frac{1}{{{{\log }_c}a}}$$     is equal to -