Question

(Faap-SP) Resolva a equação
$log_{x}(2) \times log_{x \div 16}(2) = log_{x \div 64}(2)$

Answer 1

$log_x2 \ . \ log_{\frac{x}{16}}2=log_{\frac{x}{64}}2\\\\ \frac{log_22}{log_2x}.\frac{log_22}{log_2\frac{x}{16}}=\frac{log_22}{log_2\frac{x}{64}}\\\\ \frac{1}{log_2x}.\frac{1}{log_2\frac{x}{16}}=\frac{1}{log_2\frac{x}{64}}\\\\ log_2\frac{x}{64}=(log_2x).(log_2\frac{x}{16}})\\\\ log_2x - log_264=(log_2x).(log_2x - log_216)\\\\ log_2x - 6=(log_2x).(log_2x - 4)\\\\ log_2x - 6=log^2_2x - 4log_2x\\\\ Para \ facilitar: \ log_2x=y$
$y-6=y^2-4y\\ y^2-5y+6=0\\ \Delta=(-5)^2-4.1.6=25-24=1\\\\ y=\frac{-(-5)\pm\sqrt{1}}{2}\\\\ y=\frac{5\pm1}{2}\\\\ y_1=\frac{6}{2}=3 \ e \ y_2=\frac{4}{2}=2\\\\ Agora \ temos:\\\\ log_2x=3\\ x=2^3\\ x=8\\\\ log_2x=2\\ x=2^2\\ x=4$