Question Number 208037 by hardmath last updated on 02/Jun/24
$$\mathrm{Find}:\:\:\:\mathrm{2}\:\mathrm{log}_{\sqrt{\mathrm{5}}} \:\:\mathrm{sin}\:\frac{\pi}{\mathrm{7}}\:\centerdot\:\mathrm{log}_{\sqrt{\boldsymbol{\mathrm{sin}}\:\frac{\boldsymbol{\pi}}{\mathrm{7}}}} \:\:\mathrm{5}\:\:=\:\:? \\ $$
Answered by mr W last updated on 02/Jun/24
$$=\mathrm{2}×\frac{\mathrm{log}\:\left(\mathrm{sin}\:\frac{\pi}{\mathrm{7}}\right)}{\frac{\mathrm{log}\:\mathrm{5}}{\mathrm{2}}}×\frac{\mathrm{log}\:\mathrm{5}}{\frac{\mathrm{log}\:\left(\mathrm{sin}\:\frac{\pi}{\mathrm{7}}\right)}{\mathrm{2}}} \\ $$$$=\mathrm{8}×\frac{\cancel{\mathrm{log}\:\left(\mathrm{sin}\:\frac{\pi}{\mathrm{7}}\right)}}{\cancel{\mathrm{log}\:\mathrm{5}}}×\frac{\cancel{\mathrm{log}\:\mathrm{5}}}{\cancel{\mathrm{log}\:\left(\mathrm{sin}\:\frac{\pi}{\mathrm{2}}\right)}} \\ $$$$=\mathrm{8} \\ $$
Commented by hardmath last updated on 02/Jun/24
$$\mathrm{thank}\:\mathrm{you}\:\mathrm{dear}\:\mathrm{professor} \\ $$