Найти log (36) 9 если log(36)8=m

Ответ log(36) 9 = = log36 (3^2) = = 2log(36) 3 = = 2/log(3) 36 = = 2/log(3) 9*4 = = 2/[log(3) 9 + log(3) 4] = = 2/[log(3) 3^2 + log(3) 2^2] = = 2/[2*log(3) 3 + 2*log(3) 2] = = 2/[2*{1 + log(3) 2}] = = 1/[1 + log(3) 2] (1) log(36) 8=m log(36) 2^3 = = 3*log(36) 2 = = 3/log(2) 36 = = 3/log(2) 9*4 = = 3/[log(2) 3^2 + log(2) 2^2] = = 3/[2*{log(2) 3 + lg(2) 2}] = = 3/[2*{log(2) 3 + 1}] = = 3/[2*{1/log(3) 2 + 1}] = = 3*log(3) 2 /[2*(1+log(3) 2] = m Для простоты пусть log(3) 2 = t => 3t /(2*(1+t)) = m 3t = m*2(1+t) 3t = 2m + mt 3t - mt = 2m t*(3-m) = 2m t = 2m/(3-m) В уравнение (1) 1/[1 + log(3) 2] = 1/[1 + t] = 1/[1 + 2m/(3-m)] = (3-m) /(3-m+2m) = = (3-m)/(3+m)