Wykaż że jeżeli [latex]log _{6}2 = p [/latex] oraz [latex]log _{6}5 = q [/latex] to [latex]log _{2} 6 + log _{5} 6= frac{p+q}{p*q} [/latex] Wykaż że [latex] frac{logsin30 + log sin 45+logsin60}{logcos30+logcos45+logcos60} =1[/latex]

[latex]log _{6}2 = p Rightarrow frac{1}{log _{6}2}= frac{1}{p} Rightarrow log _{2}6}= frac{1}{p}[/latex] [latex]log _{6}5 = q Rightarrow frac{1}{log _{6}5} = frac{1}{q} Rightarrow log _{5}6} = frac{1}{q}[/latex] [latex]log _{2} 6 + log _{5} 6=frac{1}{p}+frac{1}{q}=frac{q}{pq}+frac{p}{pq}= frac{p+q}{pq}[/latex] =========== [latex]frac{log sin30^o+ log sin 45^o+log sin60^o}{log cos30^o+log cos45^o+log cos60^o}=1[/latex] [latex]L=frac{log sin30^o+ log sin 45^o+log sin60^o}{log cos30^o+log cos45^o+log cos60^o}=[/latex] [latex]frac{log left(sin30^ocdot sin 45^ocdot sin60^o ight) }{log left(cos30^ocdot cos45^ocdot cos60^o ight) }=[/latex] [latex]frac{log left( frac{1}{2}cdot frac{ sqrt{2} }{2}cdot frac{ sqrt{3} }{2} ight) }{log left( frac{ sqrt{3} }{2}cdot frac{ sqrt{2} }{2}cdot frac{1}{2} ight) }=[/latex] [latex]frac{log frac{ sqrt{6} }{8} }{log frac{ sqrt{6} }{8} }=1=P[/latex]