Question

выразите log300(2700) через m и n,если m=log2(3), n=log5(2)
1)m+n/mn
2)2n/3m
3)3mn+2n+2/nm+2n+2

Answer 1

$m=log_23$

$n=log_52$

$log_{300}2700= frac{log_22700}{log_2300}=frac{log_2(2^2cdot3^3cdot5^2)}{log_2(2^2cdot3cdot5^2)}=$

$frac{log_22^2+log_23^3+log_25^2}{log_22^2+log_23+log_25^2}=frac{2log_22+3log_23+2log_25}{2log_22+log_23+2log_25}=$

$frac{2+3m+2 cdot frac{1}{log_52} }{2+m+2 cdot frac{1}{log_52} }=frac{2+3m+frac{2}{n} }{2+m+frac{2}{n} }=frac{frac{2n+3mn+2}{n} }{frac{2n+mn+2}{n} }=$

$frac{2n+3mn+2}{n}:frac{2n+mn+2}{n}=frac{2n+3mn+2}{n} cdot frac{n}{2n+mn+2} = frac{3mn+2n+2}{nm+2n+2}$

Ответ: 3) $frac{3mn+2n+2}{nm+2n+2}$