Question

log 2/3 (x) + log 3 (x) = 2
Решите пожалуйста.

Answer 1

$log_{frac{2}{3}}x+log_3x=2\\frac{log_3x}{log_3frac{2}{3}}+log_3x=2 /cdot log_3frac{2}{3}\\log_3x+log_3frac{2}{3}cdot log_3x=2log_3frac{2}{3}\\log_3x(1+log_3frac{2}{3})=log_3(frac{2}{3})^2\\log_3x(log_33+log_3frac{2}{3})=log_3frac{4}{9}\\log_3x(log_3(3cdotfrac{2}{3}))=log_3frac{4}{9}\\log_3xcdot log_32=log_3frac{4}{9} /:log_32\\log_3x=frac{log_3frac{4}{9}}{log_32}\\log_3x=log_2frac{4}{9}\\x=3^{log_2frac{4}{9}}$