Question Number 144120 by mathdanisur last updated on 21/Jun/21
$${log}_{\mathrm{2}} \mathrm{3}\:=\:{x}\:,\:{log}_{\mathrm{3}} \mathrm{5}\:=\:{y}\:,\:{lg}\mathrm{6}\:=\:? \\ $$
Answered by Ar Brandon last updated on 21/Jun/21
$$\mathrm{log}_{\mathrm{3}} \mathrm{5}=\mathrm{y}\Rightarrow\mathrm{log}_{\mathrm{2}} \mathrm{5}=\mathrm{xy} \\ $$$$\mathrm{log6}=\mathrm{log}\left(\mathrm{2}×\mathrm{3}\right) \\ $$$$\:\:\:\:\:\:\:\:\:=\mathrm{log2}+\mathrm{log3} \\ $$$$\:\:\:\:\:\:\:\:\:=\frac{\mathrm{1}+\mathrm{log}_{\mathrm{2}} \mathrm{3}}{\mathrm{log}_{\mathrm{2}} \mathrm{10}} \\ $$$$\:\:\:\:\:\:\:\:\:=\frac{\mathrm{1}+\mathrm{log}_{\mathrm{2}} \mathrm{3}}{\mathrm{1}+\mathrm{log}_{\mathrm{2}} \mathrm{5}} \\ $$$$\:\:\:\:\:\:\:\:\:=\frac{\mathrm{1}+\mathrm{x}}{\mathrm{1}+\mathrm{xy}} \\ $$
Commented by mathdanisur last updated on 21/Jun/21
$${cool}\:{Sir}\:{thankyou} \\ $$
Answered by liberty last updated on 22/Jun/21
$$\mathrm{log}\:_{\mathrm{10}} \left(\mathrm{6}\right)=\:\mathrm{log}\:_{\mathrm{10}} \left(\mathrm{2}\right)+\mathrm{log}\:_{\mathrm{10}} \left(\mathrm{3}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:\frac{\mathrm{1}}{\mathrm{1}+\mathrm{log}\:_{\mathrm{2}} \left(\mathrm{5}\right)}\:\:+\frac{\mathrm{log}\:_{\mathrm{2}} \left(\mathrm{3}\right)}{\mathrm{1}+\mathrm{log}\:_{\mathrm{2}} \left(\mathrm{5}\right)} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:\frac{\mathrm{1}+\mathrm{x}}{\mathrm{1}+\mathrm{xy}}\:. \\ $$