Question

A soma das raízes da equação log5 x . log1/5 x + 4 . log5 x = 3

Answer 1

$\displaystyle \sf \log_{5} x\cdot \log_{\frac{1}{5}}x+4\cdot \log_5x =3 \\\\ \log_5x\cdot \frac{\log_5 x}{\log_5\frac{1}{5}}+4\cdot \log_5x=3 \\\\\\ \frac{(\log_5x)^2}{\log_55^{-1}}+4\cdot \log _5x=3 \\\\\\ -(\log_5x)^2+4\cdot \log_5x-3 = 0 \\\\ (log_5x)^2-4\cdot \log_5x+3 = 0 \\\\ (\log_5x)^2-4\log_5x+4=1 \\\\\ (\log_5x - 2)^2 = 1 \\\\ log_5x-2 = \pm 1 \\\\ \log_5 x = 2\pm 1$

Daí :

$\displaystyle \sf \left\{\begin{array}{I} \displaystyle \sf \log_5x =2+1 \to log_5x = 3 \to x = 5^3 = 125 \\\\ \displaystyle \sf \log_5x=2-1\to \log_5x= 1 \to x = 5^1 = 5 \end{array} \right \\\\\\ Portanto\ o\ produto\ das\ ra{\'i}}zes\ : \\\\ 125. 5 = \boxed{ 625}\checkmark$