Question Number 18306 by mondodotto@gmail.com last updated on 18/Jul/17
Answered by diofanto last updated on 18/Jul/17
$$\frac{\mathrm{log}\:\mathrm{2}^{\mathrm{4}} \:−\:\mathrm{log}\:\mathrm{3}^{\mathrm{4}} }{\mathrm{log}\:\mathrm{3}^{\mathrm{3}} \:+\:\mathrm{log}\:\mathrm{2}^{\mathrm{3}} }\:=\:\frac{\mathrm{4}\:\mathrm{log}\:\mathrm{2}\:−\:\mathrm{4}\:\mathrm{log}\:\mathrm{3}}{\mathrm{3}\:\mathrm{log}\:\mathrm{3}\:+\:\mathrm{3}\:\mathrm{log}\:\mathrm{2}}\:= \\ $$$$=\:\frac{\mathrm{4}\left(\mathrm{log}\:\mathrm{2}\:−\:\mathrm{log}\:\mathrm{3}\right)}{\mathrm{3}\left(\mathrm{log}\:\mathrm{2}\:+\:\mathrm{log}\:\mathrm{3}\right)}\:=\:\frac{\mathrm{4}\:\mathrm{log}\:\frac{\mathrm{2}}{\mathrm{3}}}{\mathrm{3}\:\mathrm{log}\:\mathrm{6}}\:=\:\frac{\mathrm{4}}{\mathrm{3}}\mathrm{log}_{\mathrm{6}} \:\left(\frac{\mathrm{2}}{\mathrm{3}}\right) \\ $$