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

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

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

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

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