Дано [latex]log _{12} 27=a[/latex] Найти [latex]log _{6} 16[/latex]

 [latex]log_{12}3^3=a\\ log_{12}3= \frac{a}{3}\\ log_{6}16 = \frac{log_{12}16}{log_{12}6}=\frac{4log_{12}2}{log_{12}3 + log_{12}2}\\ \\ log_{3}(4*3)=log_{3}4+1=\frac{3}{a}\\ log_{3}4=\frac{3-a}{a}\\ \frac{log_{12}4}{log_{12}3}=\frac{3-a}{a}\\ log_{12}2=\frac{3-a}{6}\\ \\ log_{6}16=\frac{-4*\frac{(a-3)}{6}}{\frac{a+3}{6}}=\frac{12-4a}{a+3}\\ [/latex]

log₁₂27=3log₁₂3=3log₂3/log₂12=3log₂3/(log₂4+log₂3)=3log₂3/(2+log₂3)=a 3log₂3=2a+alog₂3⇒3log₂3-alog₂3=2a⇒log₂3(3-a)=2a⇒log₂3=2a/(3-a) log₆16=4log₆2=4log₂2/log₂6=4/(log₂2+log₂3)=4/(1+log₂3) log₆16=4/(1+2a/(3-a))=4(3-a)/(3-a+2a)=4(3-a)/(3+a)